J -P SCHNEIDERS · 2019. 5. 10. · QUASI-ABELIAN CATEGORIES AND SHEAVES Jean-Pierre Schneiders Abstract. This memoir is divided in three parts. In the first one, we introduce the

MÉMOIRES DE LA S. M. F.

JEAN-PIERRE SCHNEIDERSQuasi-abelian categories and sheavesMémoires de la S. M. F. 2e série, tome 76 (1999)<http://www.numdam.org/item?id=MSMF_1999_2_76__R3_0>

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QUASI-ABELIAN CATEGORIES AND SHEAVES

Jean-Pierre Schneiders

Abstract. — This memoir is divided in three parts. In the first one, we introduce thenotion of quasi-abelian category and link the homological algebra of these categoriesto that of their abelian envelopes. Note that quasi-abelian categories form a specialclass of non-abelian additive categories which contains in particular the category oflocally convex topological vector spaces and the category of filtered abelian groups. Inthe second part, we define what we mean by an elementary quasi-abelian category andshow that sheaves with values in such a category can be manipulated almost as easilyas sheaves of abelian groups. In particular, we establish that the Poincare-Verdierduality and the projection formula hold in this context. The third part is devoted toan application of the results obtained to the cases of filtered and topological sheaves.

Resume (Categories et faisceaux quasi-abeliens). — Ce memoire est divise en troisparties. Dans la premiere, nous introduisons la notion de categoric quasi-abelienne etrelions Palgebre homologique de ces categories a celle de leurs enveloppes abeliennes.Notons que les categories quasi-abeliennes forment une classe speciale de categoriesadditives non-abeliennes qui contient en particulier la categoric des espaces vectorielstopologiques localement convexes et la categoric des groupes abeliens filtres. Dans laseconde partie, nous definissons ce que nous entendons par categoric quasi-abelienneelementaire et montrons que les faisceaux a valeurs dans une telle categoric sontpresque aussi aises a manipuler que les faisceaux de groupes abeliens. En particulier,nous etablissons que la dualite de Poincare-Verdier et la formule de projection sontvalides dans ce contexte. La troisieme partie est consacree a une application desresultats obtenus aux cas des faisceaux filtres et topologiques.

© Memoires de la Societe Mathematique de France 76, SMF 1999

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Quasi-Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1. Quasi-abelian categories and functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1. Images, coimages and strict morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2. Definition of quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3. Strict morphisms in quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . 111.1.4. Strictly exact and coexact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.5. Exactness classes of quasi-abelian functors . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2. Derivation of quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1. The category /C(^) and its canonical t-structures . . . . . . . . . . . . . . . . . . 161.2.2. The category P(<?) and its canonical t-structures . . . . . . . . . . . . . . . . . . 191.2.3. The canonical embedding of £ in CH{£) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2.4. The category CH{£) as an abelian envelope of £ . . . . . . . . . . . . . . . . . . 31

1.3. Derivation of quasi-abelian functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.3.1. Derivable and explicitly derivable quasi-abelian functors . . . . . . . . . . . . 351.3.2. Exactness properties of derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.3.3. Abelian substitutes of quasi-abelian functors . . . . . . . . . . . . . . . . . . . . . . 431.3.4. Categories with enough projective or injective objects . . . . . . . . . . . . . . 50

1.4. Limits in quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.4.1. Product and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.4.2. Projective and inductive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651.4.3. Projective and inductive limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

1.5. Closed quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.5.1. Closed structures, rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.5.2. Induced closed structure on CH(£) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2. Sheaves with Values in Quasi-Abelian Categories . . . . . . . . . . . . . . . . . . . . 772.1. Elementary quasi-abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.1.1. Small and tiny objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.1.2. Generating and strictly generating sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.1.3. Quasi-elementary and elementary categories . . . . . . . . . . . . . . . . . . . . . . . . 822.1.4. Closed elementary categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

CONTENTS

2.2. Sheaves with values in an elementary quasi-abelian category . . . . . . . . . . . . 902.2.1. Presheaves, sheaves and the associated sheaf functor . . . . . . . . . . . . . . 902.2.2. The category of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.2.3. Internal operations on sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.3. Sheaves with values in an elementary abelian category . . . . . . . . . . . . . . . . . . 992.3.1. Poincare-Verdier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.3.2. Internal projection formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.3.3. Internal Poincare-Verdier duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.1. Filtered Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.1.1. The category of filtered abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.1.2. Separated filtered abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.1.3. The category K and filtered sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.2. Topological Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.1. The category of semi-normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.2. The category of normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1263.2.3. The category W and topological sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

MEMOIRES DE LA SMF 76

INTRODUCTION

To solve various problems of algebraic analysis, it would be very useful to have athand a good cohomological theory of sheaves with values in categories like that offiltered modules or that of locally convex topological vector spaces. The problem toestablish such a theory is twofold. A first difficulty comes form the fact that, sincethese categories are not abelian, the standard methods of homological algebra cannotbe applied in the usual way. A second complication comes from the fact that we needto find conditions under which the corresponding cohomological theories of sheavesare well-behaved. This memoir grew out of the efforts of the author to understandhow to modify the classical results in order to be able to treat such situations. Thefirst two chapters deal separately with the two parts of the problem and the last oneshows how to apply the theory developed to treat the cases of filtered and topologicalsheaves.

When we want to develop homological algebra for non abelian additive categories, afirst approach is to show that the categories at hand may be endowed with structuresof exact categories in the sense ofD. Quillen [14]. Then, using Paragraph 1.3.22 of [2],it is possible to construct the corresponding derived category and to define what isright or left derived functor.

This approach was followed by G. Laumon in [10] to obtain interesting results forfiltered ^-modules. We checked that it would also be possible to treat similarly thecase of locally convex topological vector spaces. However, when one works out thedetails, it appears that a large part of the results does not come from the particularproperties of filtered modules or locally convex topological vector spaces but insteadcome from the fact that the categories considered are exact categories of a very specialkind. In fact, they are first examples of what we call quasi-abelian categories.

To provide a firm ground for applications to other situations, we have found ituseful to devote Chapter 1 to a detailed study of the properties of these very specialexact categories.

2 INTRODUCTION

In Section 1.1, after a brief clarification of the notions of images, coimages andstrict morphisms in additive categories, we give the axioms that such a category hasto satisfy to be quasi-abelian. Next we show that a quasi-abelian category has acanonical exact structure. We conclude by giving precise definitions of the variousexactness classes of additive functors between quasi-abelian categories. This is nec-essary since various exactness properties which are equivalent for abelian categoriesbecome distinct in the quasi-abelian case.

Section 1.2 is devoted to the construction of the derived category T>(£) of a quasi-abelian category £ and its two canonical t-structures. We introduce the two corre-sponding hearts CT-L{£) and 7^%(<?) and we make a detailed study of the canonicalembedding of £ in CH(£). In particular, we show that the exact structure of £is induced by that of the abelian category CH(£) and that the derived category ofCH(£) endowed with its canonical t-structure is equivalent to P(^) endowed with itsleft t-structure. Since the two canonical t-structures are exchanged by duality, it isnot necessary to state explicitly the corresponding results for 7?.7^(<?). Note that thecanonical t-structures of V{£) and the abelian categories CH{£) and 'RT-L(£) cannotbe defined for an arbitrary exact category and give first examples of the specifics ofquasi-abelian categories. We end this section with a study of functors from a quasi-abelian category £ to an abelian category A and show that CH(£} and W-L{£) mayin some sense be considered as abelian envelopes of £.

In Section 1.3, we study how to derive an additive functor

F : £ -tT

of quasi-abelian categories. After adapting the notions of F-projective and F-injectivesubcategories to our setting, we generalize the usual criterion for F to be left or rightderivable. Next, we study various exactness properties of RF and relate them with theappropriate exactness properties of F. After having clarified how much of a functoris determined by its left or right derived functor and defined the relations of left andright equivalence for quasi-abelian functors, we show that, under mild assumptions,we can associate to F a functor

G : CU{£) -^ CH^)

which has essentially the same left or right derived functor. Loosely speaking, thecombination of this result and those of Section 1.2 shows that from the point of viewof homological algebra we do not loose any information by replacing the quasi-abeliancategory £ by the abelian category CH{£). We conclude this section by generalizingto quasi-abelian categories, the classical results on projective and injective objects.This leads us to make a careful distinction between projective (resp. injective) andstrongly projective (resp. injective) objects of £ and study how they are related withprojective and injective objects of CH{£) or 7^H(£).


INTRODUCTION 3

In Section 1.4, we deal with problems related to projective and inductive limits inquasi-abelian categories. First, we treat the case of products and show mainly thata quasi-abelian category £ has exact (resp. strongly exact) products if and only ifC1-L(£) (resp. 'RT-L(£)) has exact products and the canonical functor

£ -4- CH(£) (resp. £ -^ mi(£))

is product preserving. The case of coproducts is obtained by duality. After a detaileddiscussion of the properties of categories of projective or inductive systems of <f, wegive conditions for projective or inductive limits to be computable in £ as in CH(£).In this part, we have inspired ourselves from some methods of [4, 9]. We conclude byconsidering the special case corresponding to exact filtering inductive limits.

The last section of Chapter 1 is devoted to the special case of closed quasi-abeliancategories (i.e. quasi-abelian categories with an internal tensor product, an internalhomomorphism functor and a unit object satisfying appropriate axioms). We showmainly that in such a situation the category of modules over an internal ring is stillquasi-abelian. Examples of such categories are numerous (e.g. filtered modules overa filtered ring, normed representations of a normed algebra) but the results obtainedwill also be useful to treat module over internal rings in a more abstract category likethe category W defined in Chapter 3. We conclude by showing how a closed structureon £ may induce, under suitable conditions, a closed structure on CH(£\

In Chapter 2 we study conditions on a quasi-abelian category £ insuring that thecategory of sheaves with values in £ is almost as easily to manipulate as the categoryof abelian sheaves.

In Section 2.1, we introduce the notions of quasi-elementary and elementary quasi-abelian categories and show that such categories are very easy to manipulate. First,we study the various natural notions of smallness in quasi-abelian categories. We alsodiscuss strict generating sets which play for quasi-abelian categories the role playedusually by the generating sets for abelian categories. This allows us to introduce thedefinitions of quasi-elementary and elementary categories and to show that if £ isquasi-elementary then £H(£) is a category of functors with values in the category ofabelian groups (this an analog of Freyd's result). We also show that the category ofind-objects of a small quasi-abelian category with enough projective objects is a basicexample of an elementary quasi-abelian category. We conclude the section with a fewresults on closed elementary categories.

In Section 2.2, we show that that the category Shv(X',£) of sheaves on X withvalues in an elementary quasi-abelian category £ is well-behaved. It is even endowedwith internal operations if the category £ is itself closed. Moreover, we show that

CH{Shv(X^))wShv(X^C/H(£))

and thanks to the results in the preceding sections, we are reduced to work withsheaves in an elementary abelian category. Such sheaves where already studied in [15]

SOCIETE MATHEMATIQUE DE FRANCE 1999

4 INTRODUCTION

where it is shown that they have most of the usual properties of abelian sheaves. InSection 2.3, we give further examples of how to extend to these sheaves results whichare well known for abelian ones. In particular, we prove Poincare-Verdier duality inthis framework. We also prove that if £ is closed and satisfies some mild assumptionsthen we can establish an internal projection formula and an internal Poincare-Verdierduality formula by working almost as in the classical case.

Chapter 3 is devoted to applications to filtered and topological sheaves.In Section 3.1, we study the category of filtered abelian groups and show that this is

a closed elementary quasi-abelian with enough projective and injective objects. Its leftabelian envelope 7S is identified with the category of graded modules over the gradedring Z[T] following an idea due to Rees. We also show that the category of separatedfiltered abelian sheaves is a quasi-elementary quasi-abelian category having % as itsleft abelian envelope. Since 7^ is an elementary abelian category, the cohomologicaltheory of sheaves developed in Chapter 2 may be applied to this category and gives asatisfying theory of filtered sheaves. Since most of the results in this section are easyconsequences of the general theory, they are often given without proof.

Note that some of the results in this section where already obtained directly in spe-cific situations by various authors (e.g. Illusie, Laumon, Rees, Saito, etc.). However,to our knowledge, the fact that all the classical cohomological formulas for abeliansheaves extend to filtered abelian sheaves was not yet fully established.

It might be a good idea to read this section in parallel with Chapter 1 as it providesa simple motivating example for the abstract theory developed there.

In Section 3.2, we show first that the category of semi-normed spaces in a closedquasi-abelian category with enough projective and injective objects which has thesame left abelian envelope as the category of normed-spaces. Applying the resultsobtained before, we show that the category of ind-semi-normed spaces is a closedelementary quasi-abelian category and that its left abelian envelope W is a closed ele-mentary abelian category. We also show that the category of locally convex topologicalvector spaces may be viewed as a (non full) subcategory of W and that through thisidentification, the categories of FN (resp. DFN) spaces appear as full subcategoriesof W. Since the theory developed in Chapter 2 applies to W, we feel that W-sheavesprovides a convenient notion of topological sheaves which is suitable for applicationsin algebraic analysis. Such applications are in preparation and will appear elsewhere.

Note that, in a private discussion some time ago, C. Houzel, conjectured that acategory defined through the formula in Corollary 3.2.22 should be a good candidateto replace the category of locally convex topological vector spaces in problems dealingwith sheaves and cohomology. He also suggested the name W since he expected thiscategory to be related to the category of quotient bornological spaces introduced byWaelbroeck. We hope that the material in this paper will have convinced the readerthat his insight was well-founded.


INTRODUCTION 5

Before concluding this introduction, let us point out that discussions we had withM. Kashiwara on a first sketch of this paper lead him to a direct construction of thederived category of the category of FN (resp. DFN) spaces. These categories wereused among other tools in [8] to prove very interesting formulas for quasi-equivariantP-modules.

Note also that a study of the category of locally convex topological vector spacesalong the lines presented here is being finalized by F. Prosmans. However, in this casethe category is not elementary and one cannot treat sheaves with values in it alongthe lines of Chapter 2.

Throughout the paper, we assume the reader has a good knowledge of the theory ofcategories and of the homological algebra of abelian categories as exposed in standardreference works (e.g. [11, 12, 16] and [3, 5, 7, 17]). If someone would like anautonomous presentation of the basic facts concerning homological algebra of quasi-abelian categories, he may refer to [13] which was based on a preliminary version ofChapter 1.


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CHAPTER 1

QUASI-ABELIAN CATEGORIES

1.1. Quasi-abelian categories and functors

Let £ be an additive category with kernels and cokernels.

I.I.I. Images, coimages and strict morphisms

DEFINITION I.I.I. — Let / : E —^ F be a morphism of <f.Following [5, 16], we define the image of / to be the kernel of the canonical map

F —f Coker/. Dually, we define the coimage of / to be the cokernel of the canonicalmap Ker f —^ E.

Obviously, / induces a canonical map

Coim / —^ Im /.

In general, this map is neither a monomorphism nor an epimorphism. When it is anisomorphism, we say that / is strict

fThe following remark may help clarify the notion of strict morphism.

REMARK 1.1.2(a) For any morphism / : E —^ F of <?, the canonical morphism

Ker/ -^ E (resp. F -^ Coker/)

is a strict monomorphism (resp. epimorphism).(b) Let / : E —^ F be a strict monomorphism (resp. epimorphism) of <?. Then / is

a kernel (resp. cokernel) of

F -> Coker/ (resp. Ker/ -^ E).

(c) A morphism / of £ is strict if and only if

/ = rn o e

where m is a strict monomorphism and e is a strict epimorphism.

8 CHAPTER 1. QUASI-ABELIAN CATEGORIES

1.1.2. Definition of quasi-abelian categories

DEFINITION 1.1.3. — The category £ is quasi-abelian if it satisfies the following dualaxioms:

(QA) In a cartesian square

E-^F

E'-^F'

where / is a strict epimorphism, // is also a strict epimorphism.(QA*) In a cocartesian square

E ' - ^ F '

E-^F

where / is a strict monomorphism, // is also a strict monomorphism.

Until the end of this section, E will be assumed to be quasi-abelian.

PROPOSITION 1.1.4. — Let f : E —> F be a morphism of £. Then, in the canonicaldecomposition

of f, j is a strict epimorphism and k is a monomorphism. Moreover, for any decom-position

of f where m is a monomorphism, there is a unique morphism

h' : Coim / -> J*


1.1. QUASI-ABELIAN CATEGORIES AND FUNCTORS

making the diagram

Coim/

commutative.Dually, in the canonical decomposition

°f f ? ^* <ts an epimorphism and j* is a strict monomorphism. Moreover, for anydecomposition

of f where e is an epimorphism, there is a unique morphism

h' : I -Îm/

making the diagram

1m f

commutative.

Proof. — Let

i :Kerf-Ê

denote the canonical morphism. Since j is the cokernel of %, it is a strict epimorphism.Let us show that A* is a monomorphism. Let x : X —> Coim/ be a morphism such



that k o x = 0. Form the cartesian square

E——^Coim/T t

T 1 I Tx I rX'——;—>X

jIt follows from the fact that £ is quasi-abelian that j ' is a strict epimorphism. Since,

/ o x ' = k o j o x ' = k o x o j ' = 0,

there is a unique morphism x" : X' -^ Kerf such that i o x" = x ' . From the relationx ° f = J ° x ' = j o i o x" = 0,

it follows that x = 0.To prove the second part of the statement, note that, m being a monomorphism,

it follows from the relationm o h o i = f o i = Q ^

that h o i = 0. Since j is the cokernel of z, there is a unique morphism

h' : Coim / -^ J*

such thath' o j = h.

From the equalityk o j = m o h = m o h' o j ,

it follows thatk = m o h'.

D

COROLLARY 1.1.5. — The canonical morphism

Coim / —^ Im /

associated to a morphism f : E -^ F of £ is a bimorphism (i.e. it is both a monomor-phism and an epimorphism).

REMARK 1.1.6. — The preceding proposition shows in particular that the decompo-sition of / through Coim/ (resp. Im/) is in some sense the smallest (resp. greatest)decomposition of / as an epimorphism followed by a monomorphism. Hence, what wecall Im/ (resp. Coim/) would be called Coim/ (resp. Im/) in [12]. Despite the goodreasons for adopting MitchelPs point of view, we have chosen to stick to Grothendieck'sdefinition which is more usual in the framework of additive categories.


1.1. QUASI-ABELIAN CATEGORIES AND FUNCTORS 11

1.1.3. Strict morphisms in quasi-abelian categories

PROPOSITION 1.1.7. — The class of strict epimorphisms (resp. monomorphisms) of£ is stable by composition.

Proof. — Let u : E —^ F and v : F —^ G be two strict epimorphisms and set w = vou.We denote by iu '' Kern —^ E the canonical morphism and use similar notations forv and w. We get the commutative diagram:

Ker w ———> Ker v

Kern

One checks easily that the upper right square is cartesian. Since u is a strict epimor-phism, it follows from the axioms that k is also a strict epimorphism. To conclude,it is sufficient to prove that w is a cokernel of iw. Assume / : E —> X is a morphismsuch that f o iw = 0« Since u is the cokernel of iu and f o iu == 0, there is a uniquemorphism // : F —> X such that f o u = f. Since k is an epimorphism, the equality

/' o ̂ o k = f o iw = 0

shows that // o i^ = 0. Using the fact that v is the cokernel of z-y, we get a uniquemorphism f " : G — > X such that /// o v = f. For this morphism, we get f11 ow = fas requested. Moreover, w being an epimorphism, /// is the only morphism satisfyingthis relation. D

PROPOSITION 1.1.8. — Let

be a commutative diagram in £. Assume w is a strict epimorphism. Then, v is astrict epimorphism.

Dually, assume w is a strict monomorphism. Then, u is a strict monomorphism.

Proof. — We will use the same commutative diagram as in the proof of the precedingproposition.

SOClfiTE MATHEMATIQUE DE FRANCE 1999


First, note that the square

( u iv )EeKerv-——>F

( i o ) | \v•^- -4Ê w ' - G

is cartesian. As a matter of fact, if the morphisms

X -^ E. X -4 F

are such that

then

w o e = v o /,

v o (f — uo e) = 0

and there is h : X —^ Kerv such that

i^ o h = f — u o e.

It follows that for the morphism

(^ ) : X -> EeKerv

we have( i o ) ( ^ ) = e , (ui.)(^=/

and this is clearly the only morphism satisfying these conditions. It follows from theaxioms and the fact that w is a strict epimorphism that

( u i , ) : Ee Kerv -> F

is a strict epimorphism.Next, let x : F —)- X be such that x o i^ = 0. It follows that

x o u o i ^ = x o i ^ o k = 0

and there is x ' : G —^ X such that

Hence,

and since

{x — x' o v) o u = 0

(x — x' o v) o i^ =0

we deduce from what precedes that x = x ' o v. Since v is clearly an epimorphism,such an x ' is unique. So, v is a cokernel of iy and the conclusion follows. D



1.1.4. Strictly exact and coexact sequencesDEFINITION 1.1.9. — A null sequence

E ' ^ E -^ E"

of £ is strictly exact (resp. coexact) if e' (resp. e") is strict and if the canonicalmorphism

Ime' -^Kere"is an isomorphism. More generally, a sequence

e1 e"-1

E^ -^ ... -e—^ En {n > 3)

is strictly exact (resp. coexact) if each of the subsequences

Ei -^ E,^ -^ EW (1 < i < n - 2)

is strictly exact (resp. coexact).

REMARK 1.1.10. — It follows from the preceding definition that strict exactness andstrict coexactness are dual notions which are in general not equivalent. However, ashort sequence

0 - ^ E - ^ F -^G -^0is strictly exact if and only if u is a kernel of v and v is a cokernel of u. Hence, sucha sequence is strictly exact if and only if it is strictly coexact.

REMARK 1.1.11. — Thanks to the results in the preceding subsections, it is easilyseen that the category £ endowed with the class of short strictly exact sequencesforms an exact category in the sense of [14].

1.1.5. Exactness classes of quasi-abelian functors. — Since there are twokinds of exact sequences in a quasi-abelian category, there are more exactness classesof functors than in the abelian case. All these various classes are needed in the restof the paper. Hence, we will define them carefully in this section.

Let us first consider left exactness.

DEFINITION 1.1.12. — LetF : £ -^ T

be an additive functor.We say that F is left exact if it transforms any strictly exact sequence

0 -^ E' -^ E -^ E" -^ 0

of £ into the strictly exact sequence

0 -^ F { E ' ) -^ F(E) -^ F(E")

of T. Equivalently, F is left exact if it preserves kernels of strict morphisms.

SOClfiTfi MATHfiMATIQUE DE FRANCE 1999


We say that F is strongly left exact if it transforms any strictly exact sequence

0 -^ E1 -, E -. E"of £ into the strictly exact sequence

0 -^ F ( E ' ) -, F(E) -, F ( E " )

of T\ Equivalently, F is strongly left exact if it preserves kernels of arbitrary mor-phisms.

Finally, we say that F is regular if it transforms a strict morphism into a strictmorphism and regularizing if it transforms an arbitrary morphism into a strict mor-phism.

REMARK 1.1.13. — Some authors have defined a left exact functor between arbitraryfinitely complete categories to be a functor which preserves all finite projective limits.This definition coincides obviously with our notion of strongly left exact functor.

Other definitions of left exactness can also be introduced. They are clarified inProposition 1.1.15 the proof of which is left to the reader.

DEFINITION 1.1.14. — LetF : £ -tT

be an additive functor. Let S denote a null sequence of the form0 -^ E ' -> E -> E"

and let F(S) denote the null sequence

0 -^ F ( E ' ) -, F(E) -> F{E11).

We shall distinguish four notions of left exactness for the functor F. They are definedin the following table by the exactness property of F(S) which follows from a givenexactness property of 5.

FLL left exactLR left exactRL left exactRR left exact

Sstrictly exactstrictly exact

strictly coexactstrictly coexact

F(S)strictly exact

strictly coexactstrictly exact

strictly coexact

PROPOSITION 1.1.15. — LetF - . 8 ->T

be an additive functor between quasi-abelian categories.

(LL) The functor F is LL left exact if and only if it is strongly left exact.(LR) The functor F is LR left exact if and only if it is strongly left exact and regu-

larizing.(RL) The functor F is RL left exact if and only if it is left exact.(RR) The functor F is RR left exact if and only if it is left exact and regular.



REMARK 1.1.16. — Note also that with the notation of the preceding proposition,F is LL left exact if and only if it is RL left exact and transforms a monomorphisminto a monomorphism. Similarly, F is LR left exact if and only if it is RR left exactand transforms a monomorphism into a strict monomorphism.

Having clarified the various notions of left exactness, we can treat right exactnessby duality.

DEFINITION 1.1.17. — LetF : £ -,T

be an additive functor.We say that F is right exact if it transforms any strictly (co) exact sequence

0 ->• E' -> E -^ E" -> 0

of £ into the strictly coexact sequence

F(Ef) -^ F(E) -> F(Eff) -^ 0

of T. Equivalently, F is right exact if it preserves cokernels of strict morphisms.We say that F is strongly right exact if it transforms any strictly coexact sequence

E ' -, E -^ E" -. 0

of £ into the strictly coexact sequence

F(Et) -^ F{E) -> F{E!t) -^ 0

of F. Equivalently, F is strongly right exact if it preserves cokernels of arbitrarymorphisms.

Finally let us introduce the various classes of exact functors.

DEFINITION 1.1.18. — LetF :£ -^ T

be an additive functor.The functor F is exact if it transforms any strictly (co) exact sequence

0 -> E' -^ E ->• E" -^ 0

of £ into the strictly (co)exact sequence

0 -^ F ( E ' ) -^ F(E) -> F ( E " ) -^ 0

of F , Equivalently, F is exact if it is both left exact and right exact.The functor F is strictly exact (resp. strictly coexact) if it transforms any strictly

exact (resp. coexact) sequenceE ' -> E -^ E"

of £ into a strictly exact (resp. coexact) sequence

F ( E ' ) -^ F(E) -> F{E")

SOClfiTE MATHfiMATIQUE DE FRANCE 1999


of.F.Finally, the functor F is strongly exact if it is both strictly exact and strictly

coexact. Equivalently, F is strongly exact if it is both strongly left exact and stronglyright exact.

1.2. Derivation of quasi-abelian categories

1.2.1. The category JC(£) and its canonical t-structures. — In this subsection,we assume that £ is an additive category with kernels and cokernels. Since £ isadditive, it is well-known that the associated category JC(£) of complexes modulohomotopy is a triangulated category. Here we will show that it is also canonicallyendowed with two t-structures which are exchanged by duality.

DEFINITION 1.2.1. — A null sequence

E' -^ E -^ E"

of £ is split if, for any object X of <?, the associated sequence

Hom^X,^) -^ Hom^(X,E) -> Hom^X.E")

is an exact sequence of abelian groups. Dually, it is cosplit if, for any object X of £,the associated sequence

RomÊ'^X) -^ Hom^(E,X) -> RomÊ'.X)

is an exact sequence of abelian groups.A complex E of £ is split (resp. cosplit) in degree n if the sequence

m—l _"__v pn d v rrin+1

is split (resp. cosplit).A complex is split (resp. cosplit) if it is split (resp. cosplit) in each degree.

REMARK 1.2.2

(a) A null sequence

E ' -^ E -^ E"

of £ is split if and only if the associated short sequence

0 -^ Kere7 -> E -^ Kere" -^ 0

splits. In particular, a sequence may be split without being cosplit.(b) A complex E of £ is split if and only if it is homotopically equivalent to 0.

Hence, E is split if and only if it is cosplit.


1.2. DERIVATION OF QUASI-ABELIAN CATEGORIES 17

LEMMA 1.2.3. — Any object E of /C(<?) ma?/ be embedded in a distinguished triangle

E<Ê-> E>° ̂

of /C(<?) where E^0 is the complex

• " E ~ 2 ->E-1 -^ Ker 4-^0

(with Ker d0^ in degree 0) and ^>0 is the complex

0-^Kerd^ -^ E° -^ E1 --

(with E° in degree 0).

Proof. — Denote by i the canonical morphism from Ker (f^ to E° and let u : E^0 —Ê be the morphism defined by

( 1 for n < 0,u^ = z for n = 0,

0 for n > 0.

By definition, the mapping cone M of u is the complex:

... E-1 © IT2 -^^ Ker 4 C £;-1 -^ E° 4 E1 . . .

(with ^° in degree 0). Let a : E^ —^ M and f3 : M —^ E^ be the morphismsdefined respectively by:

( 0 for n < -1, ( 0 for n < -1,a71 = ( ^ ) for n=- l , and /^n = < ( i d ) for n = -1,

1 for n > -1, [ 1 for n > -1.

One checks easily that

f3 o a == id^>o and a o f3 — [6.M = ̂ u ° h + h o du

where h is the homotopy defined by:

.nÎ (S~01) for ^<0 .1 0 for n >0.

Therefore, E^0 is homotopically equivalent to M and the conclusion follows. D

PROPOSITION 1.2.4. — Let fC^0^) (resp. K,^0^)) denote the full subcategory of/C(<?) formed by the complexes which are split in each strictly positive (resp. strictlynegative) degree. Then, the pair

(/C^),/^0^))

defines a t-structure on /C(^).



Proof. — Thanks to the preceding lemma, we need only to prove that

Hom^(£;,F) =0

for E C /C^°(<?) and F G /C^G?). Using the preceding lemma, we get the twodistinguished triangles

E^° -, E -> E^ -^> F^° -, F -^ F>° -^ .

Thanks to Remark 1.2.2, our assumptions show that E^ ^ 0 and that F^0 ^ 0 in/C(<?) and the preceding distinguished triangles allow us to conclude that E^0 ^ Eand F ^ F^. Since one checks easily that

Hom^(^°,F>°)=0,

the proof is complete. D

DEFINITION 1.2.5. — We call the canonical t-structure studied in the precedingproposition the left t-structure o/'/C(<f). We denote by £IC{£) its heart and by LK71

the corresponding cohomology functors.

PROPOSITION 1.2.6. — The truncation functors r^, T^" for the left t-structure of/C(<?) associate respectively to a complex E the complex

_^-2 _^-i -^Ker^ -^0

(with Kerd^ in degree n) and the complex

0 -^ Kerd^-1 -^ E71-1 -> En . . .

(with En in degree n). Hence, LKn(E) is the complex

0 -^ Kerd^-1 -^ E^ -^ Kevd^ -^ 0

where Kerd^ is in degree 0.

COROLLARY 1.2.7. — The category £IC(£) is equivalent to the full subcategory of/C(<?) consisting of complexes of the form

0-^Ker / - Ê-4F-^0

(F in degree 0).

DEFINITION 1.2.8. — An object A of/X(<?) is represented by the morphism

f : E ^ F

if it is isomorphic to the complex

0-^Ker/ -^ E -4 F -^ 0

where -F is in degree 0.



REMARK 1.2.9. — Through the canonical equivalence

/C^)013 ^ 1C{£°^.the left t-structure of /C^015) gives a second t-structure on /C(<?). We call it the rightt-structure of IC{£)' We denote by 'RIC(£) its heart and by RK71 the correspondingcohomology functors. The reader will easily dualize the preceding discussion andmake the link between the right t-structure of IC{£) and cosplit sequences of £.

1.2.2. The category P(<?) and its canonical t-structures. — In this subsection,we assume that the category £ is quasi-abelian.

DEFINITION 1.2.10. — A complex E of £ is strictly exact (resp. coexact) in degree nif the sequence

rin—l ^ . 77172 ^ . r^n-\-l

is strictly exact (resp. coexact).A complex of £ is strictly exact (resp. coexact) if it is strictly exact (resp. coexact)

in each degree.

REMARK 1.2.11. — A complex of £ is strictly exact if and only it is strictly coexact.

LEMMA 1.2.12. — Two isomorphic objects of IC(£) o>re simultaneously strictly exactin degree n.

Proof. — Let £", F be two isomorphic objects of /C(<?) and assume E is strictly exactin degree n. Applying LK1^^ we see that the complexes

0 -^ Ker^-1 ^-^ ^n-1 -^-^ Ker^ -> 0

and0 -^ Ker^-1 -^ F71-1 ^^ Ker^ -> 0

are homotopically equivalent. Let

a : LTr^) -^ L^F) and f3 : L^F) -^ Ljr^)

be two morphisms of complexes such that

idL^F) -ao f3 = ho d^j<n(^) + (ILK^{F) ° h

where h is a homotopy. We know that LKn(E) is strictly exact and we have to showthat so is LK^'^F). To this end, it is sufficient to show that <^~1 is a cokernel of z^"1.

It is clear that J^T1 is an epimorphism. As a matter of fact, if g : Kerd1? —^ Xsatisfies g o S'1?"1 = 0, it follows from the relation

idKer^-Ô/?0^-^0

that g = g o a° o f3°. Since

g o a° o <%-1 = g o 6^~1 o a~1 = 0



and 8^~1 is an epimorphism, g o a° = 0 and the conclusion follows.Assume now / : F^1 —^ A is a morphism such that / o i^~1 = 0. From this

equality, we deduce that / o a-1 o i^~1 = 0. Since 6^~1 is a cokernel of ̂ -1, we geta unique morphism // : Kerd^ —)- X such that /' o 6^~1 = f o a~1. Therefore,

Since

// o f3° o <^-1 = f o <%-1 o f3-1 = f o a-1 o /T1.

idj.n-1 -a~1 o f3-1 =h°o <^-1 + ̂ -1 o h-\

we get

f - f o f3° o <^-1 = / o h° o <^-1 + / o ̂ -1 o h-\

Since / o ̂ -1 = 0, we finally get

/ = ( / / o / 3 0 + / o / ^ o ) o ^ - l

and this concludes the proof. D

REMARK 1.2.13. — As suggested by the referee, we could also prove the precedinglemma as follows. First, we note that since a split exact sequence is clearly strictlyexact, a complex which is isomorphic to zero in /C(<?) is strictly exact. Now, let E andF be two complexes which are isomorphic in /C(<?). By a well-known result of homo-logical algebra, we know that there are complexes E ' and F ' which are isomorphic tozero in /C(<?) and such that

E C E' ^ F C F '

in C ( S ) ' Using the fact that a direct sum of two complexes is strictly exact if andonly if each summand is strictly exact, the conclusion follows easily.

PROPOSITION 1.2.14. — Let

E ^ F ^ G - ^ E[l]

be a distinguished triangle of /C(<C). Assume E and G are strictly exact in degree n.Then F is also strictly exact in the same degree.

Proof. — Thanks to the preceding lemma, we may assume F is the mapping cone of

-w[-l] : G[-l] -^ E.

Hence, F^ = Ek C G^ and^ _ ( d ^ -^\_ (d^ -w"\

~ V o «% ) •(I, T?l —— I ^ ,-n I -dp=IF ~ \ 0 <%\ " /

Denote by 6 : Kerd^ C G'71"1 —^ Kerd^ the morphism induced by

( 1 —wn~ \ j-in ^, /în—1 . znn ^ r^n,n-i I . rj w (j- —> rj (DG



One checks easily that the squarecn—l°GG71-1——-——^Kerd^

fcn-i \qGn

Ker^ C G'71-1 ——^ Kerd^

is cartesian. Hence, it follows from our assumptions that 8 is a strict epimorphism.Since

^-i ^n-i _^Ker6%

is a strict epimorphism, so is

d^~1 C id^n-i : ̂ n-1 C G71-1 — Kerc% © C?71-1.

By composing with 5, we see, by Lemma 1.1.7, that^-i ^^_i _^.^

is a strict epimorphism and the proof is complete. 0

COROLLARY 1.2.15. — Strictly exact complexes form a saturated null system in /C(<?).

Proof. — The axioms for a null system are easily checked thanks to the precedingproposition. Since it is clear that a direct sum of two complexes is strictly exact ifand only if each summand is itself strictly exact, the saturation is also clear. D

DEFINITION 1.2.16. — We denote by J\f{£) the full subcategory of /C(<?) formed bythe complexes which are strictly exact. Since Af{£) is a null system, we may definethe derived category of £ by the formula:

V(£)=IC(£)/^W-A morphism of /C(<?) which has a strictly exact mapping cone is called a strict quasi-isomorphism.

LEMMA 1.2.17. — Let T be a triangulated category endowed with a t-structure

(r^.r^).Assume At is a saturated null system of T. Denote by

Q : T -> TIM

the canonical functor and by (T/Af)^0 (resp. (T/Af)^0) the essential image of Q\q-<o(resp. 0[7->o/ Then,

((TMC^cr/A/-)^)is a t-structure on T/A/' if and only if for any distinguished triangle

jCi -^ Xo -^ N -tl>

SOCIETY MATHfiMATIQUE DE FRANCE 1999


where Xi G T^, Xo G T^0 an^ N eAf, we have X^, XQ C A/'.

Proo/. — Let us proof that the condition is necessary. Consider a triangle

X^-^Xo-, N -^

of T where Xi G T^1, Xo € T^° and TV C A/". It gives rise to the triangle

Q(Xi) -^ Q{Xo) -^ Q(N) ̂

ofT/A/'. Since Q(N) ^0,

Q(Xi) ̂ Q(Xo)

is an isomorphism in T/A/'. Its inverse belongs to

Hom^(0(Xo),Q(Xi))

and our assumption shows that it is the zero morphism. Therefore, both Q(Xo} andQ(X^) are isomorphic to 0 in T / M and the conclusion follows from the fact that Afis a saturated null system.

To prove that the condition is sufficient, we have only to show that

Hom^(Q(Xo),Q(Xi))=0

for Xo e T^°, Xi G T^. A morphism from Q(Xo) to Q(Xi) is represented by adiagram

y5 / \, a

Xo Xi

where 5, a are morphisms in T, s being an A/'-quasi-isomorphism. Thus, in T, wehave a distinguished triangle

Y -^ Xo -^ N -^

where N G N . .By the properties of t-structures, we also have in T a distinguishedtriangle

YQ 4 V -^ Yi -±4

where Vo C T^0 and Vi G T^1. Let us embed s o t m a distinguished triangle

Yo -^ Xo ̂ N, ̂ .

Applying the axiom of the octahedron, we get a distinguished triangle

y^ -^ TVo — N -^ .



By our assumptions, both Vi and No are objects of A/". Therefore, t is an .A/'-quasi-isomorphism. Hence, we get the commutative diagram:

where the map from YQ to X^ is 0 since

Hom^(yo,Xi)=0.

The conclusion follows easily. D

DEFINITION 1.2.18. — Thanks to Proposition 1.2.14, the preceding lemma showsthat the left t-structure on /C(<?) induces a canonical t-structure on ?(<?), we call itthe left t-structure of V{S). We denote C7i{£) its heart and

LH" : V{£) -^ CH{£}

the corresponding cohomology functors.

PROPOSITION 1.2.19. — The truncation functors r^, r^ associated to the left t-structure of'?(<?) send a complex E respectively to

. . . _^ ̂ -2 _^ ^n-i _^ Kerd^ -^ 0

fKer d1^ in degree n) and to

0 -^ Coim^"1 -> E" -^ E^ • • .

(E71 in degree n). Hence, LH'^Ê) is the complex

0 -^ Coimd^"1 -^ Kerd^ -^ 0

(Kerd^ in degree 0).

Proof. — From the definition of the left t-structure on V(£) and Proposition 1.2.6,it is clear that r^^), T^^) are respectively canonically isomorphic to

... -^ ̂ -2 _^ ^n-i _^ Kerd^ -^ 0

(Kerd^ in degree n) and to

0 -^ Kerd^-1 -> E71-1 -^ ̂ -^ • • •

(E171 in degree n). Hence, it is sufficient to prove that this last complex is isomorphicin Z)(<?) to the complex:

0 -> Coimd^-1 -^ E71 -^ .E^1 -. • • •



(^n in degree n) through the morphism u induced by the canonical morphism

^-i ^n-i -^Coirnd^-1.

Since r^+^n) is clearly an isomorphism in P(<?), it is sufficient to show that so isT^^n)^]. This morphism is represented by the diagram:

^n-i 6n~l

0 ———> Kerd^-1 —E——> E71-1 —E——> Kerd^ ———> 0

1 ° k"1 / i I1-i- ^ J^1"1 i

0 —————^ 0 —————> Coimd^-1 -E—^ Kerd^ ———> 0

and its mapping cone is the complex

f-s-^"£;

0 ̂ Kerd^-1 -^^ En-l v ^ /) Kerd^ C Coimd^-1 (16En~l\ Kerd^ ̂ 0

(Kerd^ in degree 0). This complex is clearly strictly exact in degree —3, —2 and 0.To show that it is strictly exact in degree —1, it is sufficient to note that

and that

Coimf"^/) ^Coimd^-1

\ J E ^

r-^-1)Coimd^-1 -^—1—4 Kerd^ C Coimd^-1

is a kernel of

Kerdi C Coimd^T1 ^^n l\ Kerd^.

D

COROLLARY 1.2.20. — Let E be a complex of £. Then,

(a) The cohomology object L^Ê) vanish if and only if the complex E is strictlyexact in degree n.

(b) The complex E is an object of the category V^°(£) (resp. T>^°(£)) associatedto the left t-structure of T>(£) if and only if E is strictly exact in each strictlypositive (resp. negative) degree.

COROLLARY 1.2.21. — The left heart of £ is equivalent to the localization of the fullsubcategory of /C(<?) consisting of complexes E of the form

0 -^ Ei -8E^ Eo -> 0



(EQ in degree 0, SE monomorphism) by the multiplicative system formed by morphismsu : E —)- F such that the square

F 6F ) F^1 ————> ^Q

HI \uo

E\ ——> EQOE

is both cartesian and cocartesian.

DEFINITION 1.2.22. — An object A of CH{£) is represented by the monomorphismf : E —^ F if it is isomorphic to the complex

0 - ^ E - ^ F - ^ O

where F is in degree 0.

REMARK 1.2.23. — Through the canonical equivalence

V{£)°^ wy^),

the left t-structure of P^013) gives a second t-structure on P(<?). We call it the rightt-structure of'!)(<?). We denote by 1VH{£) its heart and by RH71 the correspondingcohomology functors. The reader will easily dualize the preceding discussion andmake the link between the right t-structure of P(^) and strictly coexact sequences of£.

1.2.3. The canonical embedding of<? in CH(£). — In this subsection, we studythe canonical embedding

£ -> LU{£}and we show that it induces an equivalence at the level of derived categories.

DEFINITION 1.2.24. — We denote

I : £ -> CH^)

the canonical functor which sends an object E of £ to the complex

0 - ^ E - ^ O

{E in degree 0) viewed as an object of CH{£).

LEMMA 1.2.25. — Assume the square

SFF^—^FoT THI \UQ

E\ —-—> EQOE

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is cocartesian. Then CokerJ^ ^ Coker <?^.

Proof, — Denote byOF : Fo —^ Cokerj^

the canonical epimorphism. It is sufficient to show that

OF o UQ : EQ —^ Coker SF

is a cokernel of S E ' Assume x : EQ —^ X is such that x o SE = 0. It follows from ourassumptions that there is x ' : FQ —^ X such that x ' o 6p == 0 and x1 o UQ = f. SinceOF is a cokernel of S p , there is x11 : Coker Sp —^ X such that x" o ep = x ' . Thereforex" o (ep ° uo) = x and x11 is clearly the only morphism satisfying this equality. D

DEFINITION 1.2.26. — Thanks to Corollary 1.2.21, the preceding lemma allows usto define a functor

c : cn{£) -^ eby sending an object A represented by the monomorphism

EÊ,

to Coker SE- For any object A of CH(£)^ we call C{A) the classical part of A.

PROPOSITION 1.2.27. — We have a canonical isomorphism

i : C o I ̂ ids

and a canonical epimorphism

e : \^cu{£} -^ I ° C .

Together, they induce the adjunction isomorphism

Hom^^(A,J(E)) =Rom,(C{A)Ê).

In particular, £ is a reflective subcategory of CH{£).

Proof. — Let E be an object of £. Since the cokernel of

0 -^ E

is clearly isomorphic to £', we get a canonical isomorphism

i(E) : C o I { E ) -Ê.

Let A be an object of CT-L^S) represented by the monomorphismn 6E^ -p-C/l ——> FJQ.

The canonical morphism

induces a morphism


EQ —^ Coker SE

e(A) :A->JoG(A) .


Since the square0———>Cokev6E

i—îis cocartesian, e(A) is an epimorphism in CT-i(£).

From the general results on adjunction formulas, we know that it is sufficient toshow that both the composition of

and

and the composition of

e(I(E)) : I ( E ) - , I o C o I ( E )

I(i(E)) : I o C o I ( E ) - ^ I ( E )

G(e(A)) :C(A) - ^ C o I o C { A )

andi(C(A)) : C o I o C ( A ) ->C(A)

give identity morphisms. This follows obviously from the definition of e and i. D

COROLLARY 1.2.28. — The canonical functor

I : S -> LU(e}

is fully faithful. Moreover, a sequence

E' -^ E -^ E"

is strictly exact in £ if and only if the sequence

I ( E ' } -^ I{E) -^ I{E11)

is exact in CT-LÊ).

Proof. — From the preceding proposition, it follows that

Horn ̂ ^(J(^)J(F)) = Rom^C o I{E), F).

Since C o I ^ id^, we see that I is fully faithful.Assume the sequence

0 -^ E' -4 E —^ E" -^ 0

is strictly exact. By a well known property of the heart of a t-structure, the cokernelof I(e1) : I ( E ' ) —^ I { E " ) is obtained by applying the functor LHQ to the complex

->/ e0 -^ E ' -4 E -^ 0

SOCIETfi MATHfiMATIQUE DE FRANCE 1999


(E in degree 0). Since e' is a monomorphism, CokerJ(e') is represented by this samecomplex. The map

0 ———> E ' —e—^ E ——> 0

1° [elt

0 ———> 0 ———> E" ———> 0being clearly a strict quasi-isomorphism, it follows that CokerJ(e') ^ I { E " ) . Fromthe adjunction formula of Proposition 1.2.27, we know that I is kernel preserving.Hence, the sequence

0 -^ I ( E ' } -^\ I(E) ̂ \ I{E"} -^ 0

is exact.Assume now that the sequence

E ' -^ E ^\ E"

is strictly exact. Applying the preceding result to the sequence

0 -^ Kere' ->£;-> Kere" -^ 0

and using the fact that I is kernel preserving we see easily that the sequence

I ( E 1 ) -, I ( E ) — I{E11)

is exact.Finally, assume the sequence

I ( E ' ) -^ I ( E ) ̂ \ I { E t t )

is exact. From what precedes, it follows that

I^Coime') =CoimI(e'),

J(Kere") =KerJ(e//).

Since CoimJ(e/) ^ KerJ(e / /), the result follows from the fact that I is fully faithful.D

PROPOSITION 1.2.29(a) An object A of CT-L{£) represented by the monomorphism

E\ —> EQ

is in the essential image of I if and only if SE is strict.(b) Assume

A-^Bis a monomorphism in CT-L(£) and B is in the essential image of I. Then A isalso in the essential image of I.



(c) Assume

0 -^ A' -^ A ̂ A7' -, 0

z'5 a 5/ior^ e:mc^ sequence in CH(£) where both A' and A" are in the essentialimage of I. Then A is also in the essential image of I.

Proof. — (a) & (b) If OE is strict, then the sequence

0 -^ Ei -^ Eo -^ Cokerj^ -^ 0

is strictly exact and by applying the functor I , we see that A ^ J(Cokerrf^).Assume now that there is an object F of £ and a monomorphism

A-^J(F).

By Proposition 1.2.27, we know that this monomorphism is induced by a morphism

C(A) — F.

Hence, the canonical morphism

A ̂ J(C7(A))

is also a monomorphism. This means, by definition, that the complex

0 -4- Ei -6E^ Eo -4 CokeroE -^ 0

is strictly exact at EQ. Therefore SE is strict and the conclusion follows.(c) Assume A is represented by the monomorphism

EÊo

and A" is isomorphic to I ( E " ) where E" is an object of £. Since the morphisma" : A —^ A" comes from a morphism C(A) —^ E11\ it is represented by a morphismof complexes

l̂-Ô

1 ° 1"^ ^0 — — — ^ E "

Since the mapping cone of this morphism is the complex

0 ̂ Ei ̂ EQ ̂ E" -> 0,

the kernel of a" is represented by the monomorphism

EI AKera.

associated to SE- By assumption, this kernel is in the essential image of J. By (a), itfollows that f3 is a strict monomorphism. Since the canonical monomorphism

Ker a —^ Eo

SOClfiTE MATHfiMATIQUE DE PRANCE 1999


is also strict, Proposition 1.1.7 shows that SE itself is a strict monomorphism and (a)allows us to conclude. Q

DEFINITION 1.2.30. — For any object A of CH(£}, we define the vanishing part ofA to be the kernel V(A) of the canonical epimorphism

e(A) :A-^ I o C ( A ) .

REMARK 1.2.31. — For any object A of CH(£}, V(A) is represented by a bimor-phism. Moreover, V(A) ^ 0 if and only if A is in the essential image of J.

PROPOSITION 1.2.32. — The canonical embedding

I : £ -^ jCH(£)

induces an equivalence of categories

V{I):V(£)-^V(£H(£)).

which exchanges the left t-structure ofv(£) with the usual t-structure of P(£^(<?)).

Proof. — By Corollary 1.2.28, we know that I transforms a strictly exact complex of£ in an exact complex of CH(£). Therefore, there is a unique functor P(7) makingthe diagram

£——I-—>£H(£)

Qe QCU{£}^ P(J) -L

V(£)——^(CH^))commutative.

Since any object A of C'H{£) may be represented by a monomorphism

J7 6E^ TTCJ\ ———> £!JQ.

of <f, it has a resolution of the form

O-^J(Ei) -Î(Eo) Â-^0.

Hence, using Proposition 1.2.29, we may apply the dual of [6, Lemma 4.6] to theessential image of I considered as a subset of ObCT-i^). This shows that for anycomplex A of C'H{S) there is a complex E of £ and a quasi-isomorphism

I(E) -^ A.

Thanks to a well-known result on derived categories, the conclusion follows fromthe fact that a complex E of S is strictly exact in a specific degree if and only ifP(J)(£') is exact in the same degree. D



1.2.4. The category CT-L{£) as an abelian envelope o { £ . — In this subsection,we show that CH(£) is in some sense an abelian envelope of £. Although we have notstated explicitly the dual results for VM(£\ we will use them freely.

DEFINITION 1.2.33. — Let A an abelian category. We denote by 7iex(£,A) (resp.Cex(£, A)) the category of right (resp. left) exact functors from £ to A.

PROPOSITION 1.2.34. — For any abelian category A, the inclusion functor

I : £ -> £H(£)is strictly exact and induces an equivalence of categories

I' : nex{CU{£),A) -^ nex{£,A).By this equivalence, exact functors correspond to strictly exact functors.

Proof. — It follows from Corollary 1.2.28 that I is strictly exact. Hence I ' is a welldefined functor. Let us prove that it is essentially surjective. Let

F :£ -f Abe a right exact functor.

The functorH° o IC{F) o r^0 : IC{£) -^ A

transforms a quasi-isomorphism of /C(^) into an isomorphism of A. As a matter offact,let

u : X ->Ybe such a quasi-isomorphism. Since Q(u) is an isomorphism in Z)(<?), so is r^Qû) ̂Q^Tû). Denote by Z the mapping cone of

rû : r<°X -^ r^Y.

By construction, Z^ = 0 for k > 0. Since Z is strictly exact, the sequence

Z~2 -^ Z~1 -^ Z° -^ 0

is strictly exact. Applying F, we get the exact sequence

F(Z-2) -^ F(Z-1) -, F{Z°) -, 0.

Hence ^(^(F)(Z)) = 0 for k ^ -1. Since the triangle

W^X) -^ ICW^Y) -^ IC(F)(Z) ̂

is distinguished in /C(-4), the long exact sequence of cohomology shows that

Ô/CÔT^)

is an isomorphism in A.It follows from the preceding discussion that there is a functor

G : V(S) -^ A



such that GoQ=Hoo|C{F)or<o. Let

F ' : Cn{£) -^ A

be the restriction of G to CH{£). Since F ' o I ^ F, it remains to show that F ' isright exact. Assume

0 -^ A' ̂ A ̂ A" -^ 0is an exact sequence in CH{£). Since we may replace A' and A by isomorphic objects,we may assume that A' = A^ = 0 for />* > 0 and that a' is induced by a morphismof IC(£) that we still denote a'. Let Z be the mapping cone of a7. By construction,Z G /C^°(<?) and we have a distinguished triangle

A ' ^ A ^ Z ^

in IC(£). Hence, A" ^ Z in ^P(<?) and

FÂ'') ^°(K;(F)(Z)).

Applying H° to the distinguished triangle

IC{F){A1) -^ /C(F)(A) -^ /C(F)(Z) ̂

of /C(A), we get the exact sequence

F\A1) -^ F'(A) -^ FÂ") ̂ 0.

Note that when F is strictly exact, /C(F)(Z) is strictly exact in any degree A; ̂ 0 andwe get the short exact sequence

0 -^ F\A') -^ F'(A) -^ F\A") -^ 0.

To see that I ' is fully faithful, it is sufficient to recall that any object A of CH{£)may be embedded in an exact sequence

J(Fi) -^ I(Eo) -Â-^0.

D

LEMMA 1.2.35. — Let £ be a full subcategory of the abelian category A. Assume £is essentially stable by subobjects (i.e. for any monomorphism

A-Ê

of A with E in £ there is E' in £ and an isomorphism A ̂ E''). Then,

(a) Any morphism of £ has a kernel and a coimage and they are computable in A.(b) A sequence

E^F^G

of £ is exact in A if and only if

Coimn ̂ Kerv

in £.



(C) If

E-^F

E ' — - ^ F 'u

is a cartesian square in £ and u is a strict epimorphism then so is u'.

Proof. — To avoid confusions, we will make use of the canonical inclusion functorJ : £ -^ A

(a) Letu : E -> F

be a morphism of <?. SinceKer J(u) -^ J(E)

is a monomorphism, there is an object K of £ and an isomorphism

KerJ(n) ^ J ( K ) .This gives us a morphism

J ( K ) -^ J(E)which is a kernel of J{u). Since £ is a full subcategory of A, this morphism is of theform J(k) where

k : K ->Eis a morphism in £. One checks easily that A; is a kernel of u in <?. Hence,

J(Keru) ̂ KerJ(^).

Since the canonical morphism

Coim J(u) -, J(F)

is a monomorphism, there is an object C of £ and an isomorphism

CoimJ(u)^J(C).Proceeding as above, we get a morphism

c : E -tCsuch that

J(c) : J(E) -> J(C)is a cokernel of

J(k) : J(Keru) -, J(E).Therefore,

c : E ->Cis a cokernel of k : Kern —^ E and C is a coimage of u. Hence,

J{Coimu) ̂ CoimJ'('u).



Parts (b) and (c) follow directly from (a). D

PROPOSITION 1.2.36. — Let £ be a quasi-abelian category and let A be an abeliancategory. Assume that the functor

J : £ - ^ A

is fully faithful and that

(a) for any monomorphismA -> J(E)

of A there is an object E' of £ and an isomorphism

A ̂ J(E'),

(b) for any object A of A, there is an epimorphism

J{E) -^ A

where E is an object of £.

Then, J extends to an equivalence of categories

CH(£) w A.

Proof. — It follows from (b) that for any complex A € T)~(A) there is a complex Eofp~(<f) and a quasi-isomorphism

J{E) -> A.

Moreover, thanks to the preceding lemma, a complex E C V~~{£) is strictly exact indegree k if and only if J(E) is exact in degree k. It follows from these facts that Jinduces an equivalence

V~(£)^V~{A)which exchanges the left t-structure of'?"(<?) with the canonical t-structure of'D~(A).In particular,

cn(£) w A.D

1.3. Derivation of quasi-abelian functors

In this section, we assume that £ and T are quasi-abelian categories and we willgive conditions for an additive functor

F : £ -,F

to be left or right derivable (Although we do note state explicitly the correspondingresults for multivariate functors, the reader will figure them out easily). We will also


1.3. DERIVATION OF QUASI-ABELIAN FUNCTORS 35

investigate to what extend -F is determined by its left and right derived functors.Finally, we will explain how to replace -F with a functor

G : CU{£) -^ CH{£)

with the same left or right derived functor.

1.3.1. Derivable and explicitly derivable quasi-abelian functors. — As inthe abelian case, we introduce the following definition.

DEFINITION 1.3.1. — LetF : £ -rF

be an additive functor and denote as usual

^:/C(^)->P(^ Q^:W^VW

the canonical functors.Assume we are given a triangulated functor

G:^^)^^^}

and a morphismg : Q ^ o > C ^ ( F ) ^ G o Q ^

Then, (G, g) is a right derived functor of F if for any other such pair (G", </), thereis a unique morphism

h : G -^ G'

making the diagram

GoQe9^

Qô>C^{F) hoQe

G'oQecommutative. The functor F is right derivable if it has a right derived functor. In thiscase, since two right derived functors of F are canonically isomorphic, we may selecta specific one. We denote such a functor RF and call it the right derived functor ofF.

Dually, assume we are given a triangulated functor

G:V-(£)^V-(^)

and a morphismg : G o Q e ->0ô/C~(F).



Then, (G, g) is a left derived functor of F if, for any other such pair (G1\ g ' ) ^ thereis a unique morphism

h: G' -^ Gmaking the diagram

G'oQe

GoQecommutative. The functor F is left derivable if it has a left derived functor. In thiscase, since two left derived functors of F are canonically isomorphic, we may select aspecific one. We denote such a functor LF and call it the left derived functor of F.

In order to give a criterion for derivability, we will adapt the usual results forabelian functors.

DEFINITION 1.3.2. — LetF : £ -^ T

be an additive functor.A full additive subcategory T of £ is F-projective if

(a) for any object E of £ there is an object P of P and a strict epimorphism

P ~-Ê.

(b) in any strictly exact sequence

0 -^ E' -> E -. E" ->• 0

of £ where E and E" are object of P, E ' is also in P.(c) for any strictly exact sequence

0 ->• E ' -^ E -4 E" -> 0

where £", E and E" are objects of P, the sequence

0 -4 F ( E ' ) -^ F{E) -^ F{E") -^ 0

is strictly exact in f\Dually, a full additive subcategory X of £ is F-injective if

(a) for any object E of £ there is an object I of Z and a strict monomorphism

E -^1.



(b) in any strictly exact sequence

0 -^ E ' -^ E -^ E" -> 0

of £ where E ' and E are object of Z, E" is also in Z.(c) for any strictly exact sequence

0 ->• E ' ->• E -^ E" ->• 0

where £", £J and E" are objects ofZ, the sequence

0 -> F(£;') — F(E) -^ F ( E " ) -, 0

is strictly exact in T'.

LEMMA 1.3.3. — Le^ V be a subset of Ob(<f). Assume that, for any object E of £,there is a strict epimorphism

P -^ Ewith P in V. Then, for any object E of C~(£) there is a quasi-isomorphism

u: P -^ E

with P in C~(P) and such that each^k . pk _^ ^k

is a strict epimorphism.

Proof. — We may restrict ourselves to the case where Ek = 0 for k > 0. To simplifythe notations, we set as usual Ek = ̂ -A;. We will proceed by induction. Assume wehave already Pje^ Uk-, d^ such that

<0 ——> Ek ——^ Ek-i EQ ——^ 0T T TUk Uk-1 UQ

I < I I0 ——> Pfc ——^ Pk-i Po ——^ 0

is a fc-quasi-isomorphism (i.e. the mapping cone is strictly exact in degree greater orequal to k). Let us form the cartesian square

df,,Efe+i—^Kerdf

T TV'\ Uk\

E'^-^Keid^

Letw : Pfc-n -> E^

SOCIfiT^ MATH6MATIQUE DE FRANCE 1999


be a strict epimorphism with Pj^+i in P and set

^4+1 = v ° w ^k+i = v 1 o w.Let us show that

<+i0 ——> Ek+i ——^ Ek • - • • • • • - Eo ——> 0

T T T^+1 ^fe ^0

I <+i I I0 ———> Pk+, -±4 P, Po ——> 0

is a (k + l)-quasi-isomorphism. It follows from the definition of the mapping coneand from the induction hypothesis that the only thing to prove is that the sequence

Uk+i \ (d^ uk \( ^+1 \ (dî Uk \l-^J I 0 -<J^fe+i ————> Ek+i e Pk —————

is strictly exact. By construction,

p^—^êp, v ° -^Ê,eP^

( v t }FT1 v ~v \ Tri /T\ D^+1 —————^ ^+1 ® Pfc

is a kernel of/rif+i ^ ^

^+1 e PA; v ° " d f e / ) Ek e Pfc-iSince w is a strict epimorphism, the conclusion follows easily.

To conclude, let us show that z^+i is a strict epimorphism. By applying LHk, itfollows from the induction hypothesis that

Uk : Kerdf -)- Kerd^

is a strict epimorphism. Therefore, v ' is also a strict epimorphism and by composition,so is Uk+i. D

PROPOSITION 1.3.4. — Let P be an F-projective subcategory of £. Then, the fullsubcategory J\f~ (P) of iC~(P) formed by strictly exact complexes is a null system andthe canonical functor

IC~(P)/Af~{P)-^T>~(£)is an equivalence of categories. Dually, let I be an F'injective subcategory of £.Then, the full subcategory M^W of K^W formed by strictly exact complexes is anull system and the canonical functor

/C-^ZV.A^CT-^P^)is an equivalence of categories.

Proof. — Thanks to Lemma 1.3.3, the proof goes as in the abelian case. D

MfiMOIRES DE LA SMF 76


PROPOSITION 1.3.5. — Let £, F be quasi-abelian categories and let

F : £ ->^

be an additive functor.

(a) Assume £ has an F-injective subcategory. Then, F has a right derived functor

RF:V^(£) -̂ (J').

(b) Dually, assume £ has an F-projective subcategory. Then, F has a left derivedfunctor

LF:V-(£)-,V-(F),

Proof, — Thanks to Lemma 1.3.3 and Proposition 1.3.4, the proof goes as in theabelian case. D

The preceding proposition is the main tool to show that a functor is derivable.However, it does not give a necessary and sufficient condition for derivability. This isthe reason of the following definition.

DEFINITION 1.3.6. — An additive functor

F : £ -,T

is explicitly right (resp. left) derivable if £ has an F-injective (resp. F-projective)subcategory.

REMARK 1.3.7. — LetF : £ -,T

be a right derivable left exact functor. Call F-acyclic an object I of £ for which thecanonical morphism

F(J) -^ RF(I)is an isomorphism and assume that for any object E of £ there is an F-acyclic objectI and a monomorphism

E -^1.Then, F-acyclic objects of £ form an F-injective subcategory and F is explicitly rightderivable.

1.3.2. Exactness properties of derived functors

PROPOSITION 1.3.8. — LetF : £ -^ F

be an additive functor of quasi-abelian categories and let T be an F-injective subcate-gory of £. Consider the right derived functor

RF:^^) -^P-^).

Then

SOCIETY MATHEMATIQUE DE FRANCE 1999


(LL) The functor RF is left exact for the left t-structures of V~^(£) and V^^) ifand only if the image by F of any monomorphism

I i ^ I o

of £ where To; Ii 0'^ objects ofT is a monomorphism of F.(LR) The functor RF is left exact for the left t-structure of V^^) and the right

t-structure of P4' (^) if and only if the image by F of any monomorphism

Ii -Îo

of £ where IQ, Ji are objects ofT is a strict monomorphism ofT'.(RL) The functor RF is left exact for the right t-structure of T)^^) and the left

t-structure V^^F).(RR) The functor RF is left exact for the right t-structure of ̂ (S) and the right

t-structure V^ (^).

Proof(LL) The condition is necessary. Let A be an object of CH{£) represented by amonomorphism

I i ^ I owhere Jo, Ii are objects ofZ. It follows that RF{A) is isomorphic to the complex

O^FW-^FW-Ô(F(Io) in degree 0). Our assumption implies that this complex is strictly exact indegree —1. Therefore, F{8) is a monomorphism in T.

The condition is also sufficient. Let E be an object of P^0^). Since

E ̂ rÊ,

we may assume that E1^ = 0 for k < —1. Replacing E by an isomorphic complex ifnecessary, we may even assume that Ek is an object of Z for any k C Z. Since E isan object ofT^0^),

E-1 -^ E°is a monomorphism of <?. Hence,

F{E~1) -^F(E°)

is a monomorphism of T andRF(E) ̂ F(E)

is an object of P^0^).(LR) Let us show that the condition is necessary. Let A be an object of CT-L{£)represented by a monomorphism

A 4/o



where To 5 Ii are objects ofZ. It follows that RF(A) is isomorphic to the complex

o^F{h)^\F(io)ô(F(7o) in degree 0). Our assumption implies that this complex is strictly coexact indegree —1. Therefore, F(6) is a strict monomorphism in .7"'.

The condition is also sufficient. Let E be an object ofp"1"^) which is strictly exactin each strictly negative degree. Replacing E by an isomorphic complex if necessary,we may assume that £^ = 0 for k < —1 and that E^ is an object ofZ for any k ^ —1.Since E is strictly exact in degree —1, the differential

E-1 -^ E°

is a monomorphism. Therefore, our hypothesis shows that

F(E-1) -^F(E°)

is a strict monomorphism in T and the complex

RF(E) ̂ F(E)

is strictly coexact in each strictly negative degree.(RL) & (RR) Let E be an object of T)^^) which is strictly coexact in each strictlynegative degree. Replacing E by an isomorphic complex if necessary, we may assumethat E^ = 0 for k < 0 and that E1^ is an object of Z for any k ^ 0. Therefore, thecomplex

RF{E) ̂ F{E)is strictly coexact in each strictly negative degree. D

REMARK 1.3.9. — One checks easily that the condition in part (LL) of the precedingproposition is equivalent to the fact that

LH~1 o RF(A) ̂ 0

for any object A of CH(£). Similarly the condition in part (LR) of the precedingproposition is equivalent to the fact that

RH~1 o RF{A) ̂ 0

for any object A of CH{£).

PROPOSITION 1.3.10. — LetF : £ -rT

be an explicitly right derivable functor of quasi-abelian categories and consider its rightderived functor

RF:V^(£) -^(J^).

Then the canonical morphism

I o F -^ LH° o R F o I (resp. I o F -> RH° o R F o I )



is an isomorphism if and only if F is RL (resp. RR) left exact.

Proof. — We consider only the RL case, since the RR case may be treated similarly.Assume that

I o F ^ LH° o R F o I .Let

0 -> E ' -^ E ->• E" -^ 0be a strict exact sequence of £ and consider the induced distinguished triangle

I { E ' ) -, I ( E ) — I ( E " ) -±4

of P"1" (E). Applying RF and passing to cohomology, we get the exact sequence

0 -^ LH° o R F o I ( E f ) -, LH° o RF o I{E) -> LH0 o RF o I { E " )

of jC7i(J7). Using our assumption, we see that the sequence

0 -> I o F ( E ' ) -^ I o F(E) -> I o F{E")

is exact in CH(T}. Therefore the sequence,

0 -^ F{E1) -^ F(E) -> F{E11}

is strictly exact in T and F is RL left exact.Conversely, assume F is RL left exact. Let Z be an F-injective subcategory of £

and let I be a resolution of an object E of £ by objects of X. The sequence

0 -^ E -> 1° -> I1

being strictly coexact in <?, our assumption shows that the sequence

0 -^ F(E) -^ F(I°) -^ F(J1)

is strictly exact in J-\ Therefore,

I o F(E) ̂ LH° o F ( I ) ̂ LH° o RF o I ( E )

as requested. D

PROPOSITION 1.3.11. — LetF : £ -^T

be an explicitly right derivable functor of quasi-abelian categories and consider its rightderived functor

RF:V+(£) —P4-^).Then(LL) The functor RF is left exact for the left t-structures o/P"^) and V^^) and

LH° o R F o I ^ I o F

if and only if F is LL left exact.



(LR) The functor RF is left exact for the left t-structure of ̂ (S) and the rightt-structure ofT^^^F) and

RH° o R F o I c ^ I o F

if and only if F is LR left exact.(RL) The functor RF is left exact for the right t-structure of P"̂ ) and the left

t-structure P4" (^F) and

LH° o R F o I ^ I o F

if and only if F is RL left exact.(RR) The functor RF is left exact for the right t-structure of 'D^~(£) and the right

t-structure P"̂ " (.77) and

RH° oRFoI ^ I o F

if and only if F is RR left exact.

Proof. — (LL) This follows from the preceding proposition and Proposition 1.3.8.(LR) From the preceding proposition and Proposition 1.1.15, the condition is clearlysufficient. Let us show that it is also necessary. By Proposition 1.3.10, we alreadyknow that F is RR left exact. Since -F transforms any strict morphism into a strictmorphism, to conclude, it is sufficient to show that F transforms any monomor-phism into a strict monomorphism. Let A be the object of C^H(E) represented by amonomorphism

-EI —> EQof £. Consider the associated distinguished triangle

Ei -^ Eo -^ A -ôfy^). Applying the functor RF and taking cohomology, we get the exact sequence

0 -^ RH° oRFo I(E^) -. RH° o RF o I{Eo) -^ RH° o RF{A)

of 7\^(J'). Hence the sequence

O^F(Ei)^HF(£;o)

is strictly coexact in F and F(SE) is a strict monomorphism in F.(RL) & (RR) This follows directly from Proposition 1.3.10. D

1.3.3. Abelian substitutes of quasi-abelian functors. — In this subsection,our aim is to show that, under suitable conditions, a functor F : E —^ T gives rise toa functor G : CH(£) —^ CT-L^) which has the same left or right derived functor.

DEFINITION 1.3.12. — Two explicitly right (resp. left) derivable quasi-abelian func-tors are right (resp. left) equivalent if their right (resp. left) derived functors areisomorphic.



PROPOSITION 1.3.13. — LetF : £ -^T

be an additive functor.Assume F is explicitly right derivable. Then F is right equivalent to an explicitly

right derivable left exact functor

F° : £ -, Twhich is unique up to isomorphism.

Dually, assume F is explicitly left derivable. Then F is left equivalent to an ex-plicitly left derivable right exact functor

FQ : £ -, T

which is unique up to isomorphism.

Proof. — Let Z be an F-injective subcategory of <f. Let E be an object of £ and letI be a right resolution of E by objects of Z. Since

RF(E) ̂ F ( I ) ,it is clear that

LH° o RF(E)is in the essential image of

i : e -^ CU{E).Set

F° = C o LH° o I .By construction, the functor F° is left exact and is isomorphic with F on Z. Therefore,Z forms an F°-injective subcategory of £ and we get

RF° ^ RF.Hence F° is right equivalent to F.

Assume now that G : £ —^ T is an explicitly right derivable left exact functorwhich is right equivalent to F. Since we have

LH° o RG{E) ̂ I o G(E)

and RF ^ RG, we see that

G(E) ̂ C o LH° o RG(E) ̂ F°

and the conclusion follows. D



PROPOSITION 1.3.14. — LetF : £ -,T

be an additive functor between quasi-abelian categories. Assume F is explicitly rightderivable. Denote Z an F-injective subcategory of £ and consider the right derivedfunctor

RF:V^(£) -^P+(^).

In order that there exists an explicitly right derivable functor

G : Cn{£) -> CU^)

and an isomorphismRF=RGoT)(I),

it is necessary and sufficient that one of the following equivalent conditions is satisfied:

(a) The functor RF is left exact with respect to the left t-structures of T>~^(£) o,ndP+CT.

(b) The functor F° is strongly left exact.

In such a case, G is right equivalent to the explicitly right derivable left exact functor

LH° o RF : CH(£) -^ CHCF\

Moreover, the restriction of this functor to £ is isomorphic to F if and only if F isstrongly left exact.

Proof. — First, let us show that conditions (a) and (b) are equivalent.(a) => (b). Let

0 -^ A' -^ A -^ A" -> 0be a short exact sequence of CH{£). From the associated distinguished triangle

RF(A') -^ RF(A) -, RF(Att) -^

and the fact that LH~1 o RF(A") ^ 0, we deduce that the sequence

0 -^ LH° o RF{A') -> LH° o RF(A) -^ LH° o RF{A")

is exact in CR{T\ Hence,

LH° o RF : Cn(£) -> CU^)

is a left exact functor. Consider now a morphism

e' : E ' -^ E

in £. Since the sequenceO-^Kere'-^-Ê

is strictly exact in £, it gives rise to an exact sequence in CH{£). Applying LH°oRF^we get the exact sequence

0 -^ LH° oRFo J(Ker e1) -^ LH0 o RF o J(£") -> LH° o RF o I(E)



of CM,(T\ Therefore, the sequence

0 -^ F°(KeTe') -^ F ° ( E ' ) -^ F°(E)

is strictly exact in T and F° is a strongly left exact functor.(b) =^ (a). Let

I I - ^ I Qbe a monomorphism of £ with both I\ and To m ̂ Since F° is strongly left exact,

F°{I,)-^F°W

is a monomorphism of T ' . The conclusion follows from the fact that F° and F coincideon Z.

Now, let us come back to the main proof.Necessity. Since RG is left exact with respect to the left t-structures of P"^) andD" .̂?7), so is RF and condition (a) is satisfied.Sufficiency. Denote G the functor

LH° o RF : CU{£) -> CH{F)

and let J denote the full additive subcategory of CH{£) formed by the objects Asuch that

L^ o RF(A) ̂ 0for any k -^ 0. It follows from condition (a) and from the long exact sequence ofcohomology associated to F that G is a left exact functor and that for any shortexact sequence

0 -> A' -> A -^ A" -^ 0

where A' and A are objects of J , A" is also an object of J and the sequence

0 -^ G(A') -^ G(A) -^ G(A / /) -^ 0

is exact in CH{F). Let A be an object of CH(£). As an object of P(<?), A isisomorphic to a complex

0 ̂ J-1 -^ 1° ̂ I1 -^ • • .

of objects of Z. It follows that A is represented by the monomorphism

J-1 ->Kerd°.

Since the squareI-1————>I°

I-1———^Kerd°is cartesian, it represents a monomorphism

A-^ J



where J is the object of CH{£} represented by the monomorphism

I-1 -^ 1°.

Since the sequence0 -> I ( I ~ 1 ) -> 1(1°) -> J -^ 0

is exact in CT-L{£) and I ( I ~ 1 ) , 1(1°) are objects of J', we see that J is also an objectof J . Together with what precedes, this shows that J is G-injective. Now, let E bean object of P"^) and let

E-^ Jbe an isomorphism where J is a complex of Z. We have

RG o P(J)(E) ^ G o I{J) ̂ F(J) ^ J?F(E)

as requested.Assume now that

G' : CH{£) -. cn^)is an explicitly right derivable functor such that

RF^RG' op(J).

It follows from this formula thatRGoD(I) ^RG' op(J).

Since P(-^) is an equivalence of categories, we see that

RG^RG1.

Therefore G' is right equivalent to G.The conclusion then follows from the definition of -F°. D

PROPOSITION 1.3.15. — LetF ' . £ - ^ F

be an additive functor of quasi-abelian categories. Assume F is explicitly left derivableand consider its left derived functor

LF:V-{£)^V-{F).

Then, there exists an explicitly left derivable functor

G : CH(£} -> CH{F)

such thatLF = LG o p(J)

and any such functor is left equivalent to the explicitly left derivable right exact functor

LH° o LF : CH^) -^ CU^).

Moreover, the restriction of this functor to £ is isomorphic to F if and only if F isregular and right exact.

SOClfiTfi MATHfiMATIQUE DE PRANCE 1999


Proof. — Let P be a F-projective subcategory of <?. It follows from the constructionof LF that it is right exact with respect to the left t-structures of V~(£) and T)~(^F).Therefore,

LH° o LF : CH{£) -^ CH(F)is a right exact functor. Denote it by G and denote Q the essential image of I\-p.Consider a short exact sequence of CH{£)

0 -^ A' -, A -, A" — 0

where A and A" are objects of Q. Since I is a fully faithful strictly exact functor,this sequence is isomorphic to the image by I of a strict exact sequence

0 -> E ' -> E -^ E" -> 0

of £ where E and E" are objects of P. It follows that E ' is an object of V andconsequently that A! is an object of Q. Moreover, since the sequence

0 -^ F ( E ' ) ->• F{E) ->• F{E") -^ 0

is strictly exact in 'J-', the sequence

0 -^ G{A') -, G{A) -^ G(A") -> 0

is exact in CT-L^F). In order to show that Q is G-projective, it is thus sufficient tonote that since any object A of CH(£) is a quotient of an object of the form I ( E )where E is an object of <?, it follows from the fact that V is -F-projective that A isalso a quotient of an object of Q. The preceding discussion shows that G is explicitlyleft derivable. Consider the functor

LF:P-(<?)^P-(^).

Since G o I(P) ^ F(P) for any object P of V ^ we get the requested isomorphism

L G o V ( I ) ^ L F .

Assume now thatG' : CU{£) -^ LU(T)

is an explicitly left derivable functor such that

LF ^LG' op(J).

It follows from this formula that

LGo^(I) ^LG' op(J).

Since P(J) is an equivalence of categories, we see that

LG^LG'.

Therefore G' is left equivalent to G.The last part of the result follows from Subsection 1.3.2 D



PROPOSITION 1.3.16. — Let £ and T be quasi-abelian categories and let

F : £ —^ T (resp. E : F -^ £)

be an explicitly left (resp. right) derivable functor. Assume that

Hom^.(F(X),y) ^ Rom^(X,E(Y))

functorially in X C £, Y G T (i.e. F is a left adjoint of E). Then,

Hom^(LF(X),y) ^ Rom^(X^RE(Y))

functorially in X G D~{£), Y G -D+(J^).

Proof. — Let V be an F-projective subcategory of £ and let Z be an E'-injectivesubcategory of F. Using the canonical morphisms

id^) -^ r^ (n C Z)

and the properties of E-injective and F-projective subcategories, one checks easilythat any morphism

u : LF(X) -^ Y

of D(J^) may be embedded in a commutative diagram of the form

LF(X) —u—> Y

va

F(P)——^1u

where

(a) P (resp. I ) comes from an object of K~(P) (resp. K^(T)),(b) the morphisms u' : F(P) — ^ I ^ a ' . Y — ^ I come from morphisms of K{F)^(c) the isomorphism F(P) -^ LF{X) comes from a quasi-isomorphism

f3:P -> X

ofK-{£).To such a diagram, we associate the unique morphism v : X —> RE(Y) making thediagram

X —u-^ RE(Y)

4 \P-Ê(I)

commutative. Note that in this diagram v ' : P —^ E(I) is obtained from u' byadjunction and that RE(Y) —^ E(I) is induced by a. We leave it to the reader to



check that v depends only on u and that the process of passing from u to v defines afunctorial morphism

Hom^(LF(X),y) ̂ Rom^(X^RE(Y)).

Proceeding dually, we get a functorial morphism

Rom^(X^RE(Y)} -^ Hom^^(LF(X),Y)

and it is easy to check that this defines an inverse of the preceding one. The conclusionfollows. D

COROLLARY 1.3.17. — In the situation of the preceding proposition,

LH° o LF : CU{£) -^ CH^)

is a left adjoint ofLH° o RE : Cn(^) -, LU(£\

Proof. — Since E is a right adjoint of F, E is strongly left exact. In particular,

RE^H^)) C D^0^)

for the left t-structure. The conclusion follows from the isomorphisms

Horn ^^{LH° o LF(X),Y) ̂ Hom^>o^(r^° o LF(X),Y)^Rom^(LF(X)^Y)

and

Horn ^^(X,LH° o RE(Y)) ̂ Rom^^X.r^ o RE(Y))^Rom^{X^RE{Y))

holding for any X er^), y<ErH(.F). D

1.3.4. Categories with enough projective or injective objects

DEFINITION 1.3.18. — An object I of £ is injective (resp. strongly injective} if thefunctor

Hom^(.,J) ̂ ^ ->Abis exact (resp. strongly exact). Equivalently, I is injective (resp. strongly injective) iffor any strict (resp. arbitrary) monomorphism

u : E -^ F

the associated mapHorn (F, I ) -^ Horn {E, I )

is surjective.Dually, an object P of £ is projective (resp. strongly projective) if the functor

Hom^(P,-) :<? -,Ab



is exact (resp. strongly exact). Equivalently, P is projective (re'sp. strongly projective)if for any strict (resp. arbitrary) epimorphism

u : E -^ F

the associated mapHom(P.E) -.Hom(P,F)

is surjective.

REMARK 1.3.19. — What we call a strongly projective object was simply called aprojective object by some authors. We have chosen to stick to our definition forcoherence with our notions of exact and strongly exact functor and also becauseprojective objects are more frequent and more useful than strongly projective ones.

DEFINITION 1.3.20. — A quasi-abelian category £ has enough projective objects iffor any object E of £ there is a strict epimorphism

P - ^ E

where P is a projective object of £.Dually, a quasi-abelian category £ has enough injective objects if for any object E

of £ there is a strict monomorphism

E-tl

where I is an injective object of £.

REMARK 1.3.21. — LetF ' . £ -rT

be an additive functor.Assume £ has enough injective objects. Then, the full subcategory Z of £ formed

by injective objects is an F-injective subcategory. In particular, F is explicitly rightderivable.

Dually, assume £ has enough projective objects. Then, the full subcategory P of£ formed by projective objects is an P-projective subcategory. In particular, F isexplicitly left derivable.

PROPOSITION 1.3.22. — Let? (resp. X) be a full additive subcategory of £. Assumethat:

a) The objects ofP (resp. X ) are projective (resp. injective) in £.b) For any object E of £, there is an object P of P (resp. I of X ) and a strict

epimorphism (resp. monomorphism)

P—Ê (resp. E-)-I).



Then the canonical functor

iC~(P) -^ V~(£) {resp. ^(T) -> ?+(<?))

is an equivalence of categories.

Proof. — We treat only the part corresponding to P. The statement for Z will followby duality.

Thanks to Lemma 1.3.3 the proof may proceed as in the abelian case and it issufficient to show that strictly exact objects of fC~(P) are isomorphic to 0. Let P bean object of IC~(P)' Assume P is strictly exact. By definition, this means that thesequences

0 -4- Kerdf -^ Pk-> Kerdf_i -. 0are strictly exact for any k G Z. If Kerdf_^ is projective in <f, the sequence splits andKerd^ is also projective. Therefore, a decreasing induction shows that the complexP is split and the conclusion follows by Remark 1.2.2. D

PROPOSITION 1.3.23. — Using the same assumptions and notations as in the pre-ceding proposition, a sequence

E' ̂ E ̂ E"

is strictly exact (resp. coexact) in £ if and only if the sequence of abelian groups

Horn (P, E1) -4- Horn (P, E) -^ Horn (P, E " )

(resp. Horn {E", I ) -^ Horn {E, I ) -^ Horn ( E ' , I ) )is exact for any P C P (resp. I C X).

Proof. — We consider only the case of P, the other one is obtained by duality. Thecondition is clearly necessary, let us prove that it is also sufficient.

We will first show that a sequence

0 -^ E ' -^ E -^ E"

is strictly exact if the sequence

0 -^ Horn (P, E ' ) -, Horn (P, E) -^ Horn (P, E " )

is exact for any P C P. Let x : X —> E be a morphism of £ such that e" o x = 0.It follows from the preceding proposition that we may find a strict exact sequence ofthe form

Pi -^ Po -4 X -, 0where Pi and Po are in P. It follows from our hypothesis that the sequence

0 -^ Horn (Pfc, E ' ) -^ Horn (P^, E) -, Horn ( P k , E " )

is exact for k e {0,1}. Therefore, there is a morphism x ' : LQ —^ E ' such thate1 o x ' = x o e. Since

e t o x f o 6 = x o e o 6 = 0 ^



it follows that x1 o 8 = 0. Hence, there is a morphism x" : X —^ E' such thatx" o e = x ' . Clearly,

e' o x" o e = e1 o x1 = x o e

and we see that e' o x" = x. Since x" is clearly the only morphism satisfying thisproperty, it follows that e' is a kernel of e" and the sequence

0 -> E ' -4 E -^ E"

is strictly exact.To conclude, it is sufficient to show that a morphism

f :E ->F

of £ is a strict epimorphism if the associated morphism

Horn (P, E) ->• Horn (P, F)

is surjective for any P G P. But this is obvious since a relation of the form

e = f o e'

where e : P —)- F is a strict epimorphism implies that / is itself a strict epimorphism.D

PROPOSITION 1.3.24(a) An object P of £ is projective if and only if I(P) is projective in CH{£).(b) The category £ has enough projective objects if and only CH{£) has enough

projective objects. Moreover, in such a case, any projective object of CH(£) isisomorphic to an object of the form I{P) where P is projective in £.

Proof. — (a) Assume P is a projective in £. Consider an epimorphism u : A —t B inCH{£) and a morphism / : J(P) —)- B. We have to show that / factors through u.Since we may replace A, B by isomorphic objects, we may assume that A and B arerespectively represented by the monomorphisms Ei —^ EQ and Fi —^ FQ and thatu comes from the morphism of complexes

SFFI ———^ FQT TUi\ UQ\' ^ '^/i ————> 21/0

We may also assume that / is represented by the morphism of complexes

OFFI—^FOT Ti "i0———>P

SOClfiTfi MATHEMATIQUE DE FRANCE 1999


Since u is an epimorphism,v ^ •P ûo 6F ̂ j?-c/O ® ^1 —————> -TO

is a strict epimorphism. It follows from the fact that P is projective in £ that thereare two morphisms (?' : P —^ EQ, (p" : P —^ Fi such that (p = UQ o ^ + 6p o ^//.Therefore the morphism of complexes

£'1 ——> EQT TT A0———^P

induces a morphism // : I ( P ) —^ A such that u o f = f.Since I is an exact functor, it follows from the adjunction formula

Hom^^(C(A),E) ^ RomÎ{E))

that C transforms a projective object of CH{£) into a projective object of <f. Thereforean object P of £ such that I(P) is projective in CH{£) is projective in £.

(b) Assume £ has enough projective objects and let A be an object of CH(£). Weknow that there is an epimorphism

I(E) -^ A

where E is an object of £. Choose a strict epimorphism in £

P -^ E

where P is projective. We know that,

Z(P) -^ I(E)

is an epimorphism in C'H{£) and that J(P) is projective. Therefore, CH(£) hasenough projective objects.

Assume now that CT-L{£) has enough projective objects and let E be an object of£. There is an epimorphism

P -^ I ( E )in CH{£) where P is a projective object in CH{£}. Since C has a right adjoint, it iscokernel preserving and transforms epimorphisms in CT-L(£) into strict epimorphismsin £. Therefore,

C(P) -^ Eis a strict epimorphism. Since we have already remarked that C(P) is a projectiveobject of <?, it is clear that £ has enough projective objects.

Since any projective object Q of CH{£) is a quotient of an object of the form J(P)where P is a projective object of <?, it is a direct summand of such and object. Itfollows from Proposition 1.2.29 and part (a) that it is itself isomorphic to the imageby I of a projective object of <?. D



REMARK 1.3.25. — Assume £ has enough projective objects. For any object E of £and any injective object I of <?, we have

HomÊ.J) ^RHom^(E,J) ^ RHom ̂ ^(J(E),J(J)).

Therefore, Ext^^ ( I ( E ) , I ( I ) ) vanish for j > 0. Nevertheless, J(J) is not in generalinjective in CH(£).

PROPOSITION 1.3.26

(a) An object J of £ is strongly injective if and only if I(J) is injective in CH(^}.(b) Assume that for any object E of £ there is a strict monomorphism

E -^ J

where J is a strongly injective object of £. Then, £ is abelian.

Proof. — (a) Let J be an object of £ and assume J(J) is injective in CH(£). Let

E -^ F

be a monomorphism in £. We know that

I ( E ) -^ I ( F )

is a monomorphism in CH(£). Therefore,

^^cnw^WJ^J)) -^ Hom^^(J(£;),/(J))

is surjective. Since the functor

I : £ -^ CH(£)is fully faithful, it follows that

Hom^(F, J) -> Hom^.(E, J )

is surjective. This shows that J is strongly injective in £.Assume now that J is a strongly injective object of £. Up to isomorphism, a

monomorphism u of CH(£) is represented by a cartesian square

E,^F,

SE OF

F ul ^ r^j\ ——> ̂ iwhose associated sequence

( 8 E \

n —. F yul ^ v ^ v {uo ~8F \ ZT-U —>• rj\ ———^ _^Q (^ ^^ ——————). ^Q

is thus strictly exact. Denote a and /3 the second and third morphism of the precedingsequence and denote 7 the canonical morphism

EQ C î —^ Coim {3.



In CK(£\ a morphism / from

Ei\ ——> EQ

to J(J) is given by a morphism

/o : EQ -, J

such that /o ° SE = 0. Denote g the morphism

F ffi F ( ^ ° \ 7^/o W r\ ————> J.

Since g o a = 0, there is a morphism g ' : Coim f3 —^ J such that

9 = 9 ' ° 7«

Since the canonical morphismCoim/3 -> Fo

is a monomorphism in £ and J is strongly inject! ve in <?, we can extend g ' into amorphism g " : FQ —^ J . Clearly, this morphism induces a morphism h from

FI^FO

to J(J) in CH{£) such that f =hou. Hence J(J) is injective in CU{£).(b) We know that for any complex E of £ there is a complex J of £ and a quasi-

isomorphismu: E —^ J

such that, for any k G Z, J"^ is a strong injective object of <? and

^ : ̂ ̂ J^

is a strict monomorphism. Let

EiÊo

be a bimorphism of £ and denote by E the associated object of CH{£\ From whatprecedes, we may find a complex J and a quasi-isomorphism

u: E -^ J

such that

HI : Ei -^ Ji, no : ̂ o —^ Joare strict monomorphisms and J^ are strong injective objects of<f. Hence, the complex

Ji -^Ker^j

is isomorphic to E and 5' is a bimorphism. Since Ji is a strong injective object, 8'has an inverse in £. It follows that E is quasi-isomorphic to 0. Therefore 8 is anisomorphism of <f. The conclusion follows easily. D


1.4. LIMITS IN QUASI-ABELIAN CATEGORIES 57

1.4. Limits in quasi-abelian categories

1.4.1. Product and direct sums. — In this subsection, we study products inquasi-abelian categories. Our results show mainly that a quasi-abelian category £has exact (resp. strongly exact) products if and only if C1-L(£) (resp. 7^H(£)) hasexact products and the canonical functor

£ -4- Cn(£) (resp. £ -^ mi(£))

is product preserving. We also give criteria for these conditions to be satisfied. Weleave it to the reader to state the dual results for direct sums.

LEMMA 1.4.1. — Let A be an additive category and let I be a small set. Then, A1

is an additive category and the canonical functor

IC(A1) -^ 1C(A)1

is an equivalence of triangulated categories.

Proof. — For any C G C(A1) and any i G I , denote d the complex of A defined bysetting

C? = (C^,

d^ = (dg),This gives us a canonical functor

C(A1) -4 C(AyC ̂ (Ci)iei

which is trivially an isomorphism of categories. Since two morphisms

f :C -> D , g : C -> D

are homotopic in C(A1) if and only if

fi'.d-^Di, g, : d -> D,

are homotopic in C(A) for every i C J, it is clear that we have a canonical isomorphismof categories

IC(A1) ̂ 1C{A)1.In A1\ we have

(A^B), =Ai^Bi.

Hence, the preceding functor exchanges the distinguished triangles of IC(A1) with thedistinguished triangles of /C (A)1. D

LEMMA 1.4.2. — Let A be an additive category with products. Then, both C {A) and}C(A) have products.



Proof. — The functor n .^ ̂ Aiei

being additive gives rise to a functor

<[I) ^(A7)-^(.4)iei

and to a functor

/c(II):/C(-4J)-^(^zeJ

By composition with the canonical equivalences

C(AY ^C(A1)

and

this gives us functorsIC(Ay -^ JC(A1)^

11 : C(A)1 -^ C(A)11 •• C(Ay -^ C[A)iein : w -^ w)id

which are easily checked to be product functors for the corresponding categories. D

PROPOSITION 1.4.3. — Let £ be a quasi-abelian category and let I be a small set.Then, £ 1 is quasi-abelian and the canonical functor

V(£1) -^ Wis an equivalence of triangulated categories which is compatible with the left and rightt-structures. In particular, we have canonical equivalences

£H(£1) w cn{£Y, nu{£1) w mi(sY.Proof. — For any morphism / : E —^ F of £1, we have

(Ker /), •==. Ker /, and (Coker /), = Coker /,.

Therefore, an object E of IC(£1) is strictly exact in degree n if and only if the complexCi is strictly exact in degree n for any i C I . The conclusion follows easily from thisfact. D

DEFINITION 1.4.4. — A quasi-abelian category £ has exact (resp. strongly exact)products if it is complete and if the functor

n:^îd

is exact (resp. strongly exact) for any small set I .



PROPOSITION 1.4.5. — Let £ be a complete quasi-abelian category. Assume £ hasenough protective objects. Then, £ has exact products.

Proof. — LetUi: E, ->• F, (i e I )

be a small family of strict epimorphisms of £. We have to prove thatip.-îp.id id

is a strict epimorphism. Letv:P-^]^Fi

idbe a strict epimorphism where P is a projective object of £. For any i e J, it followsfrom our assumptions that there is Wi : P —^ Ei such that

p^ 0 V = Ui 0 Wi.

Let w : P —> nêz î ê ̂ ̂ q1^ morphism such thatPi OW = Wi.

Clearly,

(JJ m j o w = vid )

and the conclusion follows from the fact that v is a strict epimorphism. D

PROPOSITION 1.4.6. — Assume £ is a quasi-abelian category with exact products.Then, the category T>(£) has products. Moreover, for any small set I,

n ' ' w -^ wid

is a triangulated functor which is exact for the left t-structures. It is exact for theright t-structures, if and only if products are strongly exact in £.

Proof. — Since the functor n:^îd

is exact, it gives rise to a functor

V{]]):V{£I)-^/D{£).id

By composition with the canonical equivalence

W1 w V(£1)this gives us a functor

Pî:V(£)1 -^V{£).We will show that this is a product functor for V(£).



Let E be an object of D(f) and let F be an object of V(£Y. We have to provethat the canonical morphism

(*) Horn ̂ (E, PîFi) -^ JJ Horn ̂ (E, F,)iÎ

is bijective.Let

E ^ F .

be a family of morphisms of T>{£)- We may assume that there is a strict quasi-isomorphism

s : F ->G

of /C(<?7) and a family of morphisms

g , : E -^ d (ie I )

of /C(<?) such that the morphism fi is represented by the diagram

for each i € J. Denoteg:E-^^[Gi

iei

the morphism of/C(<?) associated to the family (gi)iei- Since Pê^ ls a functor, n%ejT 5îs a strict quasi-isomorphism of /C(<?). Hence, we may define

/ : E -^ PiîF,

as the morphism of P(<?) represented by the diagram

n^ .

Since the diagram



commutes in /C(<?), we see that the diagram

^-^nw)\^ id

f\ |̂\( 4-îs commutative in P(<?). This shows that (*) is surjective.

To show that (*) is injective, we have to prove that if

/ E^]^FiÎ

is a morphism of T>{£) such that the diagram

^n^\^ iEl

0\ \Pi\ •V

Fi

commutes in Z)(<?) for every i G J, then / = 0. Let

be a diagram of /C(<?) representing /.Recall that a diagram

z ^ z ̂ s

X Yof /C(<?) where s is a strict quasi-isomorphism, represents the morphism

X^Y

of V(£) if and only if there is a commutative diagram of /C(<?) of the form

where t is a strict quasi-isomorphism.



For any i G J, we have pi o f = 0. Therefore, there is in IC(£) a commutativediagram of the form

where ti is a strict quasi-isomorphism. Denoting

z : G - ^ ] ^ Z iz:U-41î(El

the morphism of/C(<?) associated to (^)êJ^ we get, in IC(£), the commutative diagram

n%—ip.êJ zeJ

Since Pî is a functor, ["êJ ̂ is a strict quasi-isomorphism. Therefore, / = 0 inT){£).

Since product functors are always strongly left exact, the last part of the propositionis clear. Q

COROLLARY 1.4.7. — Assume £ is a quasi-abelian category with exact products.Then,

(a) The abelian category CH{£) (resp. KH(£)) is complete and the canonical functor

£ -^ CH(£) (resp. £ -^ mi(£))is product preserving.

(b) Products are exact in CH(£).(c) Products are exact in KH(£) if and only if they are strongly exact in £.

Proof. — Let (A,),ez be a family of objects of CH{£}. We know that A, may berepresented by a monomorphism

E i - ^ F ,of £. Hence, it is clear that the object

nîî

of T>(£) is isomorphic to the complex

nîp-iei iei



Products being strongly left exact functors, this complex is in CH{£). Since CH{£) isa full subcategory of T>(<?), it follows that \\^j Ai is the product of the family (A^)^jin CH(£). Using the fact that

n ^ ̂ )7 ̂ ̂ )êJ

is triangulated, we check easily that products are exact in CH{£). This proves (b).Let (Ai)iî be a family of objects of TVH{£). We know that Ai may be represented

by an epimorphismE i ^ F ,

of £. Hence, it is clear that the object FLe/ ̂ °^ ^(<c) ls isomorphic to the complex

n^nîei iei

This complex has components in degree 0 and 1. Hence, it is in V^°(£) for the rightt-structure ofP(<?). It follows that

nHom^^Â^^Homp^^nA^^Hom^^^^ff^nA.)).id iei iei

Hence,RH°(^[A,)

ieiis a product of the family (A^)^j in 1^H{£), If products are strongly exact in <?,

a^°(nA,)^nA,zeJ iî

and the exactness of products in "KH(£) follows as in the case of C1-L{£). Conversely,if products are exact in %7^(<f), a family of morphisms (nêi °f ^ gives rise to theexact sequences

I(Ei) -^ I(Fi) -> J(Coker^) -^ 0

in T^H(£) and since

IP(^) ̂ n7^) ̂ n7^0^^) -> 0iei iei iei

is exact in 7?/^(<?), one sees that products are strongly exact in <?.To conclude, it remains to note that thanks to the preceding constructions of

products in CH(£) and 7?^(<?), the last part of (a) is obvious. D

PROPOSITION 1.4.8. — Let £ be a quasi-abelian category and assume that CH{£)(resp. 1^H(£)) is complete. Then, £ is complete.



Proof. — Let us prove that £ has products if so has CH{£). Let {Ej)^j be a smallfamily of objects of <?. Let

p,:n^)^(^)j'eJ

denote the canonical projections of the product to its factors. Let

q , : C ( ] ^ I ( E , ) ) - ^ E ,jeJ

denote the morphism obtained by composing C{pj) with the canonical isomorphism

C o I ( E , ) ^ E ^

We will prove that

C7(]p(^-))jeJ

together with the projections qj form a product of the family (Ej)j^j,First, let

Xj : X -^ E, (j e J)

be a family of morphisms of <?. Denote

x':I{X)-,\[l(E,)jeJ

the unique morphism of C7{(£) such that

P j O X 1 =I(Xj).

Letx : X ^ C ( ] ^ I ( E , ) )

jeJbe the morphism obtained by composing C ( x ' ) with the canonical isomorphism

X ^ C o I ( X ) .

Clearly,Qj 0 X = X j

for j € J.

Next, let

x:X^C([[l{E,})jeJ

be a morphism such thatq j o x = 0 (j e J).



In C'H(£), let us form the cartesian square

n /(£,) -^ i o^n I(E,))jeJ j € J

u] ]l{x)

Y———„———>IW

where the first horizontal arrow is the canonical epimorphism. Since

I ( Q j ) ° y = P jit is clear that

pj o u = I(qj) ° y ° u = I(qj o x) o v = 0and we deduce that u = 0. Therefore,

I(x} o v = y o u = 0 .

Since y is an epimorphism, so is v and we get I(x) = 0. Hence, x = 0.To conclude, let us prove that £ has products if so has 7W (<?). By duality it is

equivalent to show that £ has direct sums when CH(£) has direct sums. From theadjunction formula

Hom^(A),E) ^Hom^^(A,J(E))

it follows that, for any small family (iêJ ^d anv object X of <f, we have

Hom^(G(^J(^)),X)^Hom^^(^J(E,),J(X))^ ^7

^nHom^^(J(^),J(X))^

^J]Hom^(E,,X).z€/

Hence,C(Q)I(E,))

idis a direct sum of the family (Ei)iî in <?. D

1.4.2. Projective and inductive systems. — In this subsection, we study cate-gories of projective systems of a quasi-abelian category. We leave it to the reader tostate the dual results for inductive systems.

Let £ be a quasi-abelian category and let Z be a small category. Recall that thecategory of projective systems of £ indexed by Z is the category

^z°?

of contravariant functors from Z to £.



PROPOSITION 1.4.9. — Let £ be a quasi-abelian category and let I be a small cate-gory. Then, the category of projective systems of £ indexed by I is quasi-abelian.

Proof. — Since for any morphism

u : E -^ Fof ^zop, we have

(Ker^)(z) ^ KerH(z),(Coker^)(z) ^ Coker'u(z).

the conclusion follows easily. D

DEFINITION 1.4.10. — Let F be an object of £ and let i be an object ofZ.Assuming £ is complete, we denote F^ the object of £xop defined by setting

F\i1) = ̂ HOm^M/) = TT F.QGHom {i,i')

A projective system isomorphic to a system of the form Fi will be said to be ofelementary type.

Similarly, assuming £ is cocomplete, we denote Fi the object of ^zop defined bysetting

Fi(i') = F^011^^)) = ^ pâGHom {i' ,i)

A projective system isomorphic to a system of the form Fi will be said to be ofcoelementary type.

REMARK 1.4.11. — With the notation of the preceding proposition, Fi is exchangedwith Fi if one exchanges £ with ôp and I with Z°P.

PROPOSITION 1.4.12. — If £ is complete, then

Rom^p(E,Fi) ̂ Hom^(E(z),F)

and there is a strict monomorphism

E-^ÊWiei

for any object E of ̂ zop.Similarly, if £ is cocomplete, then

Hom^p(Fi,E) ̂ Rom^F,E(i))

and there is a strict epimorphism

(Ê(i)iÊiÎ

for any object E of £xov.



Proof. — Thanks to the preceding remark, it is sufficient to prove the first statement.For any i ' C Z, we have

Rom^(E(il),Fi(il)) ̂ Horn ̂ (Horn { i , i ' ) , Horn ( E ( i ' ) , F } ) .

These isomorphisms give us the isomorphism

Hom^.zop(E,F') ^ Hom^zf^,/^ o E)

where /i^ and ^F denotes respectively the functors

Hom(?,*) and Hom(-,F).

Using standard results on representable functors, we get

Hom^zop (E, F1) ̂ F o E(i) ̂ Horn (E(z), F).

Now let us prove the second part of the result. For any i G J, the identity morphism

E(z) -^ £'(z)

induces, by an isomorphism of the preceding kind, a morphism

E -^ E(i)1.

Together, these morphisms give us a canonical morphism

u : E -,]^[E(iY.iÎ

Choose i ' € I . To conclude, we have to prove that u(i') is a strict monomorphism.Note that (n Eaw) ̂ n E(Z)V) ^ n n E(^^

iel iGl iGZaGHom {i,i')

Composing u(i') with this isomorphism, we get a morphism

v{z') : E ( i ' ) ̂ n n E(z)zCZaeHom (^,^ / )

such thatPa OPi o v ( ^ ' ) = E{a)

for any a : i —^ i ' in Z. For a = id^/ this shows that v{i') is a strict monomorphismand the conclusion follows. D

REMARK 1.4.13. — Taking E to be a constant functor in the preceding propositionshows that

^mFî'^Fi'^l

if £ is complete.

SOCIETfi MATHEMATIQUE DE FRANCE 1999


COROLLARY 1.4.14. — Let £ be a complete (resp. cocomplete) quasi-abelian cate-gory.

(a) Assume F is an injective (resp. a projective) object of £. Then, for any i G X,F1 (resp. Fi) is injective (resp. projective) in ^zop.

(b) Assume £ has enough injective (resp. projective) objects. Then E10^ has enoughinjective (resp. projective) objects

PROPOSITION 1.4.15. — Let £ be a quasi-abelian category with exact direct sumsand let X be a small category. Then, there is a canonical equivalence

cu^^^cu^.Proof. — The canonical inclusion functor

I : £ -^ CH(£)

gives rise to a fully faithful functorI : ̂ op -^ CH^.

Let E be an object of ^zop and let

A -, I ( E )

be a monomorphism of CH^)^. Since

A(i) -> I(E(i))

is a monomorphism of C'T-L{£) for any i G Z, we can find for any i G Z an object E ' ( i )of £ and an isomorphism

I{E'{i))Â{i).Using these isomorphisms, we may turn E' into an object of ^zop such that

I ( E ' ) ̂ A.

Therefore, T^^0?) is a full subcategory of CH^)^ which is essentially stable bysubobjects. Let A be an object of /^((f)2'01'. We know that there is a canonicalepimorphism

QA(^Â.id

Since, for any i G Z, there is an epimorphism

I(E(i)) -^ A(z)

with E{i) in <f, we get an epimorphism

Î(E(i))Â.id

By definition,

(Î(E(i)W)=@ ^ Wi))Î{@ @ E{i))iel iÎ a-.i'—î iCi a-.i'—^-i



Therefore,

®wîs an object of ^(^zop). Applying Proposition 1.2.36, we get the conclusion. D

1.4.3. Projective and inductive limits

PROPOSITION 1.4.16. — Let £ be a quasi-abelian category and let I be a small cat-egory. Assume S has exact products. Then,

I(^mE{i))^^mI(E{i))z€Z iel

in CU{£). Moreover, a similar formula holds in 1^H(£) if and only if the functor

^m:£1 -,£iÎ

is regular.

Proof. — The first part follows directly from the fact that I has a left adjoint. Letus now consider the second part.The condition is sufficient. Denote J the full stfbcategory of S1- formed by thefunctors E for which the canonical morphism

I(^mE(i)) -^^mI(E{i))iei »ez

is an isomorphism in 1VI-L{£). Thanks to Corollary 1.4.7

i : s -^ nn(£)preserves products. Hence, for any object F of £ and any i' e Z, we have

J(F^)) = I(FY\i)

for any i G I and it follows from Remark 1.4.13 that F1' is an object of J . Sinceone checks also easily that a product of objects of J is in J , it follows from Propo-sition 1.4.12 that any objet E of £1 may be embedded in a strictly coexact sequenceof the form

0 -^ E -^ J° -^ J1

where J0 and J1 are in J . For such a sequence, the sequence

0 - ^ I o E - ^ I o J ° - ^ I o J 1

is exact in 7^7^ (f)^ Projective limit functors being strongly left exact, the sequence

0 -^ }^mI(E(i)) -^ ̂ mI(J°(i)) -^ ̂ 1(^(1))id iEl iel

is exact in 7?/H(<?) and the sequence

0-^^mE(i) -^^mj°(i) -^^mj1^)iel iel id



is strictly exact in £. Thanks to our assumption, the last morphism in this secondsequence is strict. Hence the sequence

0 — I(^mE(i)) -, I(^mJ°{i)) -, I(^mJ\i))iÎ iel i€l

is exact in TiH{£). The conclusion follows easily.The condition is necessary. Let

f :E ->F

be a strict morphism of £1. Since the sequence

0 - ^ K e r f - ^ E - ^ F

is strictly coexact in £x, the sequence

0 -^ I o Kerf -^ I oE -> I o F

is exact in K'H(£)1. Hence, the sequence

0 -^ ^mJ(Ker/(z)) -> 3^mJ(E(z)) -,^mI(F(i))iÎ iÎ id

is exact in 7?/H(<f). Thanks to our assumption, it follows that the sequence

0 -^ I(^mKerf(i)) -> I(^mE(i)) -> I(^mF{i))iei iei iei

is also exact in 7iT-i(£). Hence, the sequence

0 -^ ̂ mKerf{i) -^ ]^mE(i) -^ ^mF(z)i€l iÎ i€l

is strictly coexact in £ and the morphism

]^mE(i) -^^mF(i)zcz iei

is strict. D

PROPOSITION 1.4.17. — Let £ be a cocomplete quasi-abelian category. Then filteringinductive limits are exact in CH{£} and commutes with

I : £ -^ CU{£)

if and only if filtering inductive limits are strongly exact in £.

Proof. — The condition is easily seen to be necessary, let us prove that it is also suf-ficient. A strongly exact functor being regular, the dual of the preceding propositionshows already that filtering inductive limits commute with I . To prove that filteringinductive limits are exact in CH{£)^ note that the functor

Hm : £1 -^ £iei


1.5. CLOSED QUASI-ABELIAN CATEGORIES 71

being strongly exact, it is strictly exact. Hence, by Proposition 1.2.34, it gives rise toan exact functor

L : Cn(£1) -^ CH(£}and a canonical isomorphism

L o Iî Îô lira .zez

Composing L with the canonical equivalence

(*) CU^Sf w CH(£1)

we get an exact functorL' :£H(£f -^£n(£).

Let A be an object of CH{£)1. Equivalence (*) shows that, in CH^S)1, we have anexact sequence of the form

0 -^ Is o Ei -^ Ie o Eo -^ A -> 0

where(5 : EI -^ Eo

is a monomorphism of ^z. It follows that the sequence

0 -^ L o J^z(Ei) -^ L o I^x(Eo) -^ L'{A) -^ 0

is exact in CT-i(£). Therefore, the sequence

0 ̂ lnn^(Ei(z)) ̂ liin^(£;o(^)) ̂ L\A) -^ 0iez êz

is exact in CH{£). It follows that

L'(A) ^lnnA(z)zez

and one checks easily that this isomorphism is both canonical and functorial. Theconclusion follows directly. Q

1.5. Closed quasi-abelian categories

1.5.1. Closed structures, rings and modules. — Recall (see e.g. [11]) that aclosed additive category is an additive category £ endowed with an internal tensorproduct

T : £ x £ -t £,

a unit objectU G Ob(<?),

an internal homomorphism functor

H : e^ x £ -, £



and functorial isomorphisms

T(E, F) ̂ T(F, E)

T(U,E)ÊHom^(r(E,F),G) ^ Hom^^G));

these data being subject to a few natural coherence axioms.Let £ be a closed additive category. By a ring in £, we mean an unital monoid of

£. It corresponds to the data of an object R of £, a multiplication morphism

m : T(R, R) -^ Rand a unit morphism

u : U -^ R.These data being assumed to give rise to the usual commutative diagrams expressingthat m is associative and that u is a unit for m.

Let R be a ring in £. By an ^-module, we mean an object E of £ endowed with a(left) action of R. This action is a morphism

a:T(R,M) -^ Mwhich gives rise to the usual commutative diagrams expressing its compatibility withthe multiplication m and the unit u of R.

A morphism of an JZ-module E to an Ji-module F is defined as a morphism

f : E - ^ F

of £ which is compatible with the actions of E and F. One checks easily that, withthis definition of morphisms, fi-modules form an additive category which we denoteby Mod(R).

We leave it to the reader to check the following result.

PROPOSITION 1.5.1. — Let £ be a closed additive category and let R be a ring of£. Assume £ is quasi-abelian (resp. abelian). Then, Mod(R) is quasi-abelian (resp.abelian). Moreover, the forgetful functor

Mod(R) -^ £

preserves limits and colimits. In particular, a morphism of Mod(R) is strict if andonly if it is strict as a morphism of £.

PROPOSITION 1.5.2. — For any object E of £, the multiplication m of R inducesan action of R on

T(RÊ).For any R-module F , we have

^^ModW^WE)^) ̂ HomÊ,F).

In particular, T(R,E) is projective in Mod(R) ifE is projective in £.



Proof. — The fact that T{R^ E) is an -R-module is obvious. Let us prove the isomor-phism. Define

^ : Hom^^(T(^£;),F) -> Hom^F)by setting

(p(h) = ho awhere a : E —^ T(R, E) is the composition

E ^ T ( U ^ E ) ^'^WE)

and^ : Hom^F) ̂ Hom^^(r(Ê),F)

by setting^(/i)=aôr(id^/i)

where ap is the action of R on J7'. A simple computation shows that ^ is an inverseof (p and the conclusion follows. D

1.5.2. Induced closed structure on CT-LÊ)PROPOSITION 1.5.3. — Let £ be a closed quasi-abelian category with enough projec-tive objects. Denote

T : £ x £ -,£

the internal tensor product, U the unit object and

H : <?°P x £ -^ £

the internal homomorphism functor. Assume that for any projective object P thefunctor T(P, •) is exact and that T(P,P') is projective if P1 is projective. Then,

(a) H{P^') is exact if P is projective,(b) H(',I) is exact if I is injective,(c) H(P,I) is injective if P is projective and I is injective.

Moreover, T is explicitly left derivable, H is explicitly right derivable and we have thecanonical functorial isomorphisms

(d) LT(X^Y)^LT(Y^X),(e) LT(XÛ)^X^LT(U^X),(f) RHom (LT(X, Y), Z) ̂ RHom (X, RH(Y, Z)),(g) R H { U ^ Z ) ^ Z .

where X , Y C ?-(<?), Z C TÔ?).

Proof. — Thanks to our assumptions, (a), (b) and (c) follow directly from the ad-junction formula

Horn (T(X, V), Z) = Horn {X, H(Y, Z)).

Let P denote the full subcategory of £ formed by projective objects. It follows fromthe hypothesis that (P,£) is T-projective and that (P°^,£) is 7:f-injective. Therefore,



T is explicitly left derivable and H is explicitly right derivable. To prove (d), (e)and (f), we may reduce to the case where X, Y are objects of IC~CP)' In this case,LT(X, Y) ̂ T(X, Y) and RH{Y, Z) = H(Y, Z). Since T(X, Y) is an object oflC~(P),everything follows from the fact that £ is a closed quasi-abelian category. To prove(g), we use (f) with Y = (7. This gives us the isomorphism

RHom (X, Z) ̂ RHom (X, RH(U, Z))

where X G P~(<?), Z G P"^). Fix Z C Z^G?) and denote G the cone of thecanonical morphism

Z -,H(U,Z) ~^RH(U,Z).It follows from what precedes that

RHom (X, C) ̂ 0

for any X G V~{£). Hence, the complex

Hom(X.G)

is exact for any X C V and C itself is strictly exact (see Proposition 1.3.23). Therefore,C ^ 0 and

Z ^RH{U,Z).D

COROLLARY 1.5.4. — In the situation of the preceding proposition, £7i(£) is canon-ically a closed abelian category. Its internal tensor product is given by

f = LH° o LT : CH(£) x CH(£} -^ Cn(£),

its unit object by U == I(U) and its internal homomorphism functor by

H = LH° o RH : Cn^)01^ x CH(C} — CH(C}.

The functor f (resp. H) is explicitly left (resp. right) derivable and we have thecanonical isomorphisms

L T ( I ( X ) , I ( Y ) ) ^LT(X,V)

andR H ( I ( Y ) ^ I ( Z ) ) ^ R H ( Y ^ Z )

for any X , Y e P~(<?) and any Z € P" )̂.Assume moreover that the functor

T(P, • ) : e -^ eis strongly exact for any projective object P of S. Then, for any protective object Qof jC'H(S) the functor

T(Q^}:CH(£)-^Cn{£)

is exact and T(Q^Q') is projective if Q' is projective in CH(£).

M^MOIRES DE LA SMF 76


Proof. — It follows from parts (d) and (e) of the preceding proposition that T issymmetric and that I{U) is a unit. The coherence axioms are also easily checked.

Let V, Z e CH(£). We know that Y is isomorphic to a complex P of K,~(P).Therefore,

R H ( Y , Z ) ^ H ( P , Z ) .The first non-zero component of H(P, Z) is of degree -1 and is isomorphic to

H(P°,Z-1),

the second non-zero component is of degree 0 and isomorphic to

Horn (P°, Z°) C Horn (P-1, Z-1)

the differential beingfRom(P^d^)\\Rom(dp\Z-1)}'

Since Z G CH{£), d^1 is a monomorphism. So, the differential of degree -1 ofH ( P , Z ) is also a monomorphism and RH(Y,Z) G P^°(<f) for the left t-structure.Moreover, we have also LT(X,Y) G P^°(<f) for any X, Y C £^(<f). Therefore, usingpart (e) of the preceding proposition, we get successively

Hom cnw (W Y)^) ̂ Horn ̂ ^ (r^0 o LT(X, Y), Z)

^Hom^(LT(X,y),Z)

^Hom^^(X,^^(y,Z))

^ Horn ^ô(^) (X, T^° o ÂT(y, Z))

^Hom^^(X,^(y,Z))

for x, y, z e £^(<?).Since CH{£) has enough projective objects, T is clearly left derivable.Consider P C P and an exact sequence

0 -^ A7 -> A -^ A" -^ 0

of CU{£). This sequence corresponds to a distinguished triangle

A' -^ A -, A" ̂

of ?+(<?). Hence,

RH(P,A') -^ RH(P,A) — RH(P,A") -^

is a distinguished triangle of ?+(<?). Since Jf(P, •) is strongly left exact,

RH(P,Y)^H(P,Y)

is in CH(£) when Y is replaced by A', A or A". Therefore, the sequence

0 ̂ H(I(P),A1) -^ H{I(P),A) -^ ̂ (P),^) -, 0



is exact in CH(£). It follows that (J(P), CH{£)) is ^-injective and that H is explicitlyright derivable.

To conclude, it is sufficient to recall that any object X of P~(<?) is isomorphic tothe image of an object of ]C~(P) and to note that if X, Y e P and Z e £, we have

L t { I ( X ) J ( Y ) ) ^ t ( I ( X ) ^ I ( Y ) )^LH°oLT(X,Y)^T(X,V)

and

R H ( I ( Y ) ^ I ( Z ) ) ^ H ( I ( Y ) ^ I ( Z } }^ LH° o RH(Y, Z)^H(Y^ZY

Let us now treat the last part of the statement. Thanks to Proposition 1.3.24,we may assume that Q = I(P) and that Q' = I ( P ' ) where P and P ' are projectiveobjects of £. Let

0 -^ A' -^ A -^ A" -> 0be an exact sequence of CM,(£\ This corresponds to a distinguished triangle

A' -^ A -, A11 -ôfp~(<f). Applying LT(J(P), •), we get the distinguished triangle

LT(I(P),A') -^ LT(I(P),A) -> LT(I(P),A") -^We know that

LT(J(P),X)^r(P,X)for any object X of P~(<f). Therefore, the long exact sequence of cohomology showsthat r(J(P),.) is exact on CU{£) if

LH~1{T{P,AII)} ^0for any object A" of CH{£\ This will clearly be the case if

T(P, •) : £ -, £

preserves monomorphisms. Keeping in mind the fact that T(P, •) is exact and stronglyright exact, this last condition is equivalent to the one in our statement. To concludewe only have to note that

t ( I ( P ) ^ I { P I ) ) ^ I { T { P ^ P I ) )

and use Proposition 1.3.24. D


CHAPTER 2

SHEAVES WITH VALUES INQUASI-ABELIAN CATEGORIES

2.1. Elementary quasi-abelian categories

2.1.1. Small and tiny objects. — In this subsection, we study various notions ofsmallness for an object of a quasi-abelian category. We leave it to the reader to statethe corresponding results for the dual notions.

DEFINITION 2.1.1. — An object E of a cocomplete additive category £ is(a) small if

Horn (£;, ̂ F,) ̂ (9 Horn (£;, F,),i<El iÎ

for any small family (J^)^j of £.(b) tiny if

hm Horn (E, F(i)) ̂ Horn (E, Inn F(i))iei iî

for any filtering inductive system

E : I -^ £.

REMARK 2.1.2. — Here, we have followed Grothendieck's definition of a small ob-ject. We don't know if the notion of a tiny object was already defined before. Ofcourse, every tiny object is small. It is also easy to see that in a abelian category,every small projective object is tiny but this is not necessarily the case in a quasi-abelian one. There is also a possible stronger condition of smallness. This conditionis clarified in the following proposition.

PROPOSITION 2.1.3. — Let £ be a cocomplete quasi-abelian category. An object E of£ is such that

Horn (E, lim F(j)) ^ hm Horn (£;, F(j))j e j j e j

for any inductive systemF : J -, £

if and only if E is a small strongly projective object of £.

78 CHAPTER 2. SHEAVES WITH VALUES IN QUASI-ABELIAN CATEGORIES

Proof. — Taking for Z a discrete category or a category of the form

^^ ^ ^ ^J)one sees that

Horn (£;,.)preserves direct sums and cokernels. Hence, E is a small strongly projective object of<?.

Conversely, if E is a small strongly projective object, Hom(E,.) is both cokerneland direct sum preserving. Viewing

Inn F(j)3^J

as the cokernel of the canonical morphism

9 F ( j ) ^ @ F ( j )a:j—^j' 3^J

allows us to conclude. D

PROPOSITION 2.1.4. — Let £ be a cocomplete quasi-abelian category. Assume fil-tering inductive limits are exact in £. Then, a small projective object of £ is tiny.

Proof. — Let Z be a small filtering category and let P be a small projective objectof E. Denote Q the full subcategory of S1 formed by the functors E for which thecanonical morphism

Inn Horn {P,E(i)) -^ Hom(P,limE(z))z€Z iÎ

is an isomorphism. Since P is small, it is clear that Q is stable by direct sums.Moreover, for any E in £ and any i C Z, we have

Horn (P, Ei{i')) ̂ Horn (P, Q) E) ̂ Q) Hom(P.E) ^ Rom(P,E)i(i1)a-.i—^-i' a:i—î'

and it follows from the dual of Remark 1.4.13 that the functor Ei belongs to Q.Therefore, using Proposition 1.4.12, we see that any object E of £1 may be embeddedin a strictly exact sequence of the form

Qi -^ Qo -^ E -^ 0

where Qi and Qo belong to Q. Since filtering inductive limits are exact in <?, thesequence

limQi(z) —^ }imQo(i) —^ \imE(i) —^ 0iÎ iel iÎ

is strictly exact in £. Since P is projective in ^, we see that the sequences

Horn (P, Qi (z)) -^ Horn (P, Qo(i)) -> Horn (P, E(i)) -^0 (i € I )

Hom(P,limQi(z)) -^ Horn (P, Inn Qo(i)) -> Horn (P.hm E(i)) -> 0iÎ i^T iel


2.1. ELEMENTARY QUASI-ABELIAN CATEGORIES 79

are exact. Inductive limits being exact in the category of abelian groups, it followsthat

Inn Horn (P,E(z)) ^ Horn (P, Inn E(i)).iel i€l

D

2.1.2. Generating and strictly generating sets. — Although, in this subsec-tion, we consider only generating and strictly generating sets, the reader will easilyobtain by duality similar considerations for cogenerating and strictly cogeneratingsets.

Let us recall (see e.g. [11]) that a subset Q of Ob(<?) is a generating set of £ if forany pair

/E———^Ff

of distinct parallel morphisms of <f, there is a morphism

G -Ê

with G C G, such that/ o e ̂ /' o e.

It is clearly equivalent to ask that for any strict monomorphism s : S —^ E which isnot an isomorphism, there is a morphism

G -tE

with G G Q which does not factor through s. Moreover, if £ is cocomplete and G issmall, it is also equivalent to ask that for any object E of £ there is an epimorphismof the form

@G,Êj'eJ

where (Gj)j^j is a small family of elements of GThe preceding notion is suitable for the study of abelian categories where any

monomorphism is strict. For quasi-abelian ones, the following definition is moreuseful.

DEFINITION 2.1.5. — A strictly generating set of £ is a subset G of Ob(£) such thatfor any monomorphism

m : S -fEof £ which is not an isomorphism, there is a morphism

G-Ê

with G G G which does not factor through m.



LEMMA 2.1.6. — Letu :E —f F be a morphism of £. Then, the following conditionsare equivalent:

(a) u is a strict epimorphism,(b) u does not factor through a monomorphism s : S —^ E which is not an isomor-

phism.

Proof. —(a) =^ (b). Let s : S —^ E be a monomorphism. Assume u' : S —^ E is such that

u = s o u'.

Since u is a strict epimorphism, so is s. Hence, s is an isomorphism and the conclusionfollows.(b) =^ (a). Since u factors through the monomorphism Coimu —^ E^

Coim u ̂ E

and u is a strict epimorphism.D

PROPOSITION 2.1.7. — Let £ be a cocomplete quasi-abelian category. A small subsetG of Ob(<?) is a strictly generating set of £ if and only if for any object E of £, thereis a strict epimorphism of the form

@G,ÊjeJ

where { G j ) j ^ j is a small family of elements of Q.

Proof. — Assume Q is a strictly generating set of<f. Consider the canonical morphism

(D G ^ E •G'G^,/iGHom(G,^)

Using the preceding lemma, let us prove by contradiction that e is a strict epimor-phism. Let

m : S -)- Ebe a monomorphism which is not an isomorphism and assume e = m o f for some/ : X —^ S. For any G G G and any h G Horn (G, E), we get

h=mo f oi^G^h)'

in contradiction with Definition 2.1.5.Conversely, assume we have a strict epimorphism

@G,Êj^J

and let m : S —> E be a monomorphism which is not an isomorphism. Assume thatfor any j ^. J

ho Sj : Gj —^ E



factors through m. This gives us a family of morphisms

^ : G, -^ S {j e J)such that

h o Sj = m o //..

Consider the morphismh^.^G^S

j'eJassociated to the family (h'j)j^j. Clearly,

m o h1 = h.

This contradicts the fact that h is a strict epimorphism. Therefore, one of the

h o Sj

does not factor through m and Q is a strict generating set of £. D

PROPOSITION 2.1.8. — Let G be a small strictly generating set of the cocompletequasi-abelian category £. Then, a sequence

0 -^ E ' -^ E -^ E"

is strictly exact in £ if and only if the sequence of abelian groups

0 -> Horn (G, E') -^ Horn (G, E) -^ Horn (G, E")

is exact for any G € Q.Assume moreover that the elements of Q are projective objects of £. Then, a

sequence

E' -4 E -^ E"

is strictly exact in £ if and only if the sequence of abelian groups

Horn (G, E ' ) -^ Horn (G, E) -^ Horn (G, E " )

is exact for any G G G.

Proof. — Proceed as in the proof of Proposition 1.3.23. D

PROPOSITION 2.1.9. — Let £ be a cocomplete quasi-abelian category. Assume £ hasa small strictly generating set Q of small (resp. tiny) objects. Then, direct sums (resp.filtering inductive limits) are strongly exact in £.

Proof. — Let0 -^ E' -^ E -^ E"

be a strict exact sequence of £1\ where Z is a small discrete (resp. filtering) category.For any i C Z, the sequence

0 -^ E\i) -^ E{i) -^ E"(z}

SOCIETE MATHfiMATIQUE DE FRANCE 1999


is strictly exact in £. Hence, for any G e 0, we get the exact sequence

0 -> Horn (G, E'{i)} -^ Horn (G, E(i)) -^ Horn (G, ̂ (z))

By taking the inductive limit and using the fact that G is small (resp. tiny), we getthe exact sequence

0 -^ Horn (G, Inn E^)) -^ Horn (G, Inn E(z}} -^ Hon^G.lmÊ^z))iGZ ^€2: iel

The conclusion follows easily from Proposition 2.1.8. D

2.1.3. Quasi-elementary and elementary categoriesDEFINITION 2.1.10. — A quasi-abelian category is quasi-elementary (resp. elemen-tary) if it is cocomplete and has a small strictly generating set of small (resp. tiny)projective objects.

REMARK 2.1.11. — One checks easily that an abelian category is elementary if andonly if it is quasi-elementary. So, the preceding definition is compatible with thedefinition of elementary abelian categories in [15].

PROPOSITION 2.1.12. — A quasi-abelian category £ is quasi-elementary if and onlyif CH{£) is elementary.

Proof. — Let us prove that the condition is necessary. Thanks to the dual of Proposi-tion 2.1.9, direct sums are strongly exact in £. Therefore, the dual of Proposition 1.4.7shows that CT-LÊ) is cocomplete and that

I : £ -^ jCn(£)

preserves direct sums. Let P be a strictly generating small set of small projectiveobjects of £. Clearly, I(P) is a generating small set of CT-L{£) and Proposition 1.3.24shows that the objects of I(P) are projective in CH{£). To check that I ( P ) is smallfor any object P of P, we may proceed as follows. Let (A^)^j be a family of CH{£).We get a family of short exact sequences of CH{£)

0 -^ I(Fi) -^ I{E,) -^ A, -^ 0.

Therefore, the sequence

9J(F,)^Q^(^)-^®AÔiel iÎ iÎ

is exact in C1-L{£) and the sequence

Horn (J(P), ̂ J(F,)) -^ Horn (J(P), (f) I(E,)) -^ Horn (J(P), ̂ A,) ̂ 0id id id

is exact in A b. Since(Î(E,)=IÊ,)iei iî



and P is small, we get the exact sequence

(D Horn (P, F,) -^ (]) Horn (P, E,) -^ Horn (I(P)^ A,) -^ 0.êJ zeJ I(EZ

It follows thatQ) Horn (J(P), A,) ^ Horn (J(P), ̂ A,)^ iei

and we see that I ( P ) is small in CH{£}.The condition is also sufficient. Since £1-i(£) is cocomplete, it follows from the

dual of Proposition 1.4.8 that £ is cocomplete. Let P be a generating small set ofsmall projective objects of CH{£). It follows that C(P) is a generating small set ofprojective objects of <?. To check that C(P) is small for any object P of P, we canproceed as follows. Since the canonical morphism

Î(E,)-Î(Q)E,)id i€l

is epimorphic, we get an epimorphism

Horn (P,(D I(E,)) -^ Horn (P, I(Q) E,)).iEl iÎ

Since P is small, we see that the canonical morphism

Q) Horn (P, I(Ei)) -^ Horn (P, J(^ E,))i(El iei

is surjective. By adjunction, it follows that the canonical morphism

(3) Horn (C(P), Ei) -, Horn (C7(P), ̂ E,)iGJ zeJ

is surjective. Since it is also clearly injective, C(P) is small. D

REMARK 2.1.13. — Let A be a cocomplete abelian category and let P be a fulladditive subcategory of A. Assume that the objects of P form a strictly generatingsmall set of small projective objects of A. Then, thanks to a result of Freyd (seee.g. [12, Theorem 5.3]) the functor

h:A-Âdd(POP,Ab)A \-^ Hom^(.,A)

induces an equivalence of categories. Hence, we could view the following propositionas a corollary of the preceding one. However, for the reader's convenience, we preferto give a direct proof.

PROPOSITION 2.1.14. — Let£ be a cocomplete quasi-abelian category and let? be afull additive subcategory of£. Assume that the objects ofP form a strictly generating



small set of small protective objects of £. Then, the functor

h : £ -, Add^.Ab)î-^Hom^(.,E)

is strictly exact and induces an equivalence of categories

CH{£)wAdd{V°^,AbY

Proof. — For any strictly exact sequence

E ' -4 E -4 E"

of £ and any object P of P^ the sequence

Hom^P.E") -^ Hom^(P.E) -^ Hom^P.E")

is exact since P is projective. Hence, the functor h is strictly exact and induces afunctor

h:V-(£)^V-(Add(POP,Ab)).Consider the category C whose objects are denned by

Ob(£) = {{Pi)iî : I small set, P, e Ob(P)}

and whose morphisms are denned by

Hom^((P,)^(Pêj) = nOHo"^^)-id jeJ

So, a morphism / ofHom^((P,),ez,(Pn,ej)-••-•-^"^VV-1 i /zfc-f '> \ j ^

may be considered as an infinite matrix (fji) with

/„ : Pi -^ PJ Vz C J, Vj G J.

the set

U : fjz + 0}being finite for any i C J.

Now, let us define the functorS : C -^ £

by setting5((Pêz)=OPz

zeJfor any object (Pêz OI ^' For any morphism

we define/: [pi)iei -^ (P^J

5(/):QP.^9^'.iî jeJ



by settingS(f)osi=^s',of^

jeJwhere

^:P^®P. ^^ /^®^^/ j-ej

are the canonical morphisms.Since Pi is small for any i G I , the functor 5 is fully faithful. As a matter of fact,

for any objects (Pê/ and (Ppj-eJ of £, we have successively

Hom,(5((P,)^), S((P^j)) ̂ Hom,(9 P, ̂ P^)iî jeJ

^nO110111^ '̂)i(El j^J

^Hom^((P,)^(Pj),ej).Hence, C is equivalent to a full subcategory S{£) of <?. Since the direct sum of afamily of projective objects is a projective object, the objects of S(£) are projective.Moreover, by hypothesis, for any object E of <f, there is a strict epimorphism

S((Pî) -^ Ewhere (PêJ is an object of £. Therefore, by Proposition 1.3.22 we have an equiva-lence of categories

JC~(S(C))wV-(£),and the functor S induces an equivalence of categories

IC~W^V-(£).

Let P be an object of P. Recall that the functor^ : pop ̂ ̂

is a small projective object of AcM(/POP,A6). As a matter of fact, for any object Fo{Add(P°^,Ab), we have

Hom^.F)^^?).Hence, for any family (F)î of Add^P^.Ab) we get

Hom(^,^F,)^(^F,)(P)iÎ iEl

^®W)iei

^^Hom^.F,),iÎ

and h13 is small. Moreover, if

0 -^ F ' —^ F -^ F " -^ 0

SOCIETfi MATHEMATIQUE DE FRANCE 1999


is an exact sequence of Add(POP,Ab) then the sequence

0 -^ F\P) -^ F(P) -. F " ( P ) -> 0

is exact. Therefore, the sequence

0 -^ Horn (^p, F'} -, Horn (^p, F) -^ Horn (^p, F") -^ 0

is also exact and hp is projective.For any object F of ^4^(P°P,.46), we define the morphism

v : ^ ^P ̂ F{(PJ):P€^,/€F(P)}

by setting

VOS^f) =^p(f)

where

^?(/) : ̂ -^ Fis defined by

^p{f)(P'){g)=F{g)(f)

for any object P1 of P and any morphism ^ of Horn ̂ (?',?). Let us show that v isan epimorphism. It is sufficient to show that for any object P ' of P the morphism

v(P') : (f) /^(P7) -> ^(P7){(PJ):PCP, /GP(P)}

is surjective. Consider /' G ^(P7). Since idp/ G ^p/ (P') = Hom^(P /,P /), we have

^(P^^Xidp/)) = ̂ (/^(PQ^dp/)

-F(idp/)(n

=f1

and the conclusion follows.Let

S ' : C -^ Add^.Ab)

be the functor defined by setting

^((P^)^®/^.êJ

Thanks to the preceding discussion, we may apply to S ' the same kind of argumentswe applied to S and conclude that 5' induces an equivalence of categories

S ' :JC~(C) ^V-Âdd^.Ab)).



Moreover, since in the diagram

we have clearly h o S = S ' , it follows that

h:V-{£) ^V-ÂddiP^.Ab))

is also an equivalence of categories.Using the fact that h is strictly exact, we see that it exchanges the hearts of T^~ (£)

and VÂdd^P^.Ab)) corresponding left t-structures. So, h induces an equivalenceof categories

h: CU{£) -^ CHÂdd^.Ab)).Since the category Add^^Âb) is abelian,

CnÂdd^.Ab)) w AddiP^.Ab)

and the proof is complete. D

PROPOSITION 2.1.15. — Let £ be a quasi-abelian category. Assume £ is quasi-elementary. Then,

(a) Both the categories £ and CH{£) are complete with exact products. Moreover,

I :£ -^ CH(£)

preserves projective limits.(b) Both the categories £ and CH(£) are cocomplete with strongly exact direct sums.

Moreover,I : £ -^ CH(£)

preserves direct sums.(c) Both the categories £ and £H(£) have enough projective objects. Moreover,

CH{£) has enough injective objects.

Proof(a) It follows from the preceding proposition that C'H(£) is complete. Hence, Propo-sition 1.4.8 shows that £ is complete. Thanks to Proposition 1.4.5, £ has exactproducts. Hence, the conclusion follows from Corollary 1.4.7.(b) This follows from Proposition 2.1.12.(c) Obvious. D

PROPOSITION 2.1.16. — Let £ be a quasi-abelian category. Assume £ is quasi-elementary. Then, £ is elementary if and only if one of the following equivalentconditions is satisfied



(a) The functor

I : £ -^ CU{£)

preserves filtering inductive limits.(b) Filtering inductive limits are exact in £.(c) Filtering inductive limits are strongly exact in £.

Proof. — This is a direct consequence of Proposition 2.1.4, Proposition 2.1.9 andProposition 1.4.17. D

PROPOSITION 2.1.17. — Let £ be a small quasi-abelian category with enough pro-jective objects. Then,

Tnd{£)

is an elementary quasi-abelian category and there is a canonical equivalence of cate-gories

CU(Xnd{£))wlnd{CU(£}).

Proof. — Proceeding as in the abelian case (see [15]) and using well-known resultson ind-objects (see e.g. [1]), one can check that the category Tnd{£) is an elemen-tary quasi-abelian category. Denote V the full additive subcategory of £ formed byprojective objects. It follows from Proposition 2.1.14 that

CU(Tnd{£)) w Add^.Ab)

Since any object of CH(£) is a quotient of an object of I ( P ) and since any such objectis projective, Xnd{CT-L{£}} is an elementary abelian category and

Xnd(£H(£)) w Add(V°^,Ab).


2.1.4. Closed elementary categoriesPROPOSITION 2.1.18. — Let £ be a closed quasi-abelian category with T as internaltensor product and let R be a unital ring in £.

(a) Assume P is a small (resp. tiny) object of £. Then, T(R,P) is a small (resp.tiny) object of M.od(K).

(b) Assume Q is a strictly generating set of £. Then,

{T(R,G):GeQ}is a strictly generating set of Mod(R).

(c) Assume £ is quasi-elementary (resp. elementary). Then Mod{R) is quasi-elementary (resp. elementary).



Proof. — Part (a) follows directly from Proposition 1.5.2.To prove (b), let E be an fi-module and let s : S —^ E be a monomorphism

of Mod(R) which is not an isomorphism. Since Q is a strictly generating set of £,Definition 2.1.5 shows that there is G —^ E in £ which cannot be factorized throughs in £. It follows that the associated morphism T(R,G) —)- E of Mod{R) cannot befactorized through s in Mod(R) and we get the conclusion.

Part (c) is a direct consequence of (a) and (b). D

PROPOSITION 2.1.19. — Let £ be a small quasi-abelian category with enough pro-jective objects. Assume £ is endowed with a closed structure with T as internal tensorproduct, H as internal homomorphism functor and U as unit object. Then, there is aclosed structure on Xnd(£) which extends that of £ and any two such extensions arecanonically isomorphic. Assume moreover that for any projective object P of £ thefunctor

T(P, •) : £ -> £is exact (resp. strongly exact) and that T(P^P') is projective for any projective objectP' of£. Then, similar properties hold for projective objects ofXnd(£).

Proof. — Assume Xnd(£) is endowed with a closed structure extending that of £.Denote T ' its internal tensor product, H ' its internal homomorphism functor and U 'its unit object. Using the canonical fully faithful functor

"•" :£ -,Xnd(£)

we may express the fact that the closed structure of Xnd(£) extends that of £ by theformulas

U ' ^ 'V\ r'C'FVF") ^ 'T(F,F)", H ' ^ E ' ^ F ' ) ̂ "H(E,FY\

Let E : X —^ £, F : J —^ £ and G : /C —^ £ be three filtering inductive systems of £.It follows from the adjunction formula between T ' and H ' that

T\\Ê{iy\ hm ̂ (jD ̂ Inn Inn 'T(^), F(j)YiEi j e j i e i j e j

and that^/(llm"F(Jy^l"$"G(A;r) ^ l^m lim^FO')^^))".

JEJ fee/C jeJk^K,

These formulas show directly that T ' and H ' are unique up to canonical isomorphism.We may also use them to construct a closed structure on Xnd(£) extending that of £(details are left to the reader).

Since any projective object ofXnd(£) is a direct factor of a projective object of theform

®îÎ

with Pi projective in <?, the last part of the statement is clear. D

SOCI6T6 MATHfiMATIQUE DE FRANCE 1999


2.2. Sheaves with values in an elementary quasi-abelian category

In this section, we will fix a small topological space X and an elementary quasi-abelian category £ and we will show that the category of sheaves with values in £ canbe manipulated almost as easily as the category of abelian sheaves.

2.2.1. Presheaves, sheaves and the associated sheaf functor

DEFINITION 2.2.1. — A presheaf on X with values in £ (or S-presheaf) is a functor

F : Op(X)op -, £

where Op{X) denotes the category of open subsets of X with the inclusion maps asmorphisms. If V C U are two open subsets of X, we denote

rû : F(U) -^ F(V)

the associated restriction morphism. We define the category of presheaves on X withvalues in £ by setting

Psh{X^)=£OPWOP.

Let F be an object of Psh(X\ £). We define the fiber Fy; of F at x G X by setting

F,= hm F(V)V(EV^

where Vx denotes the set of open neighborhoods of x ordered by inclusion.For any U 6 Vx, we denote

rû '' F(U) -^ F,

the canonical morphism.For any open subset U of X and any ^-presheaf F, we denote F\jj the <?-presheaf

obtained by restricting the functor F to Op(U)°^.An object F of Psh(X',£) is a mono-presheaf if for any open subset U of X and

any covering V of [7, the morphism

r : F(U) -^ J] F(V)v^v

defined by setting pv o r = r^ ̂ is monomorphic. Equivalently, F is a mono-presheafif and only if HE o F is an abelian mono-presheaf for any object E of £.

An object F of Vsh{X\£) is a sheaf (or £-sheaf) if for any open subset U of Xand any covering V of U we get the strict exact sequence

0 -^ F(U) ̂ JJ F(V) ̂ JJ F(W H W)v^v w,w'ev

where r is defined as above and

Pw,w o r ' = r^rw/ ,w °Pw - r^wnw ,w ° Pw •Equivalently, F is a sheaf if and only if HE ° F is an abelian for any object E of £.


2.2. SHEAVES WITH VALUES IN AN ELEMENTARY QUASI-ABELIAN CATEGORY 91

We denote by Shv(X',£) the full subcategory ofPsh{X,£) formed by sheaves.

PROPOSITION 2.2.2. — The category Psh(X',£) is a quasi-abelian category and

CH(Psh{X^£)) ̂ Psh(X',CH(£)).

Moreover, if P is a strictly generating full additive subcategory of small projectiveobjects of £. Then, the canonical functor

h : Psh(X^£) -^ Add(P,Psh(X',Ab))

which associates to an £-presheaf F the functor

P^hpoF

factors through an equivalence of categories

CH(Psh(X^£)) -^ Add(P^Psh(X',Ab)).

In particular, h is a fully faithful strictly exact functor.

Proof. — The first part of the result follows from Proposition 1.4.9. Since £ has exactdirect sums, it follows from Proposition 1.4.15 that

Cn{Psh(X^£)) ̂ Psh{X',£H(£)).

Since Proposition 2.1.14 shows that

CH(£)Âdd(P,Ab),the conclusion follows easily. Q

DEFINITION 2.2.3. — Let V be a covering of X. We define L(V; F) to be the kernelof the morphism

Y[F(V)^ JJ F{Wr}W).vev w.w'ev

We set alsoL(X;F)= hm L(V;F)

VeCv{X)°P

where Cv{X) denotes the set of open coverings of X ordered by setting V < V if forany V C V there is V G V such that V C V.

Finally, we define the <?-presheaf L(F) by setting

L{F)(U)=L(U^F\u).We have a canonical morphism

F->L(F).

PROPOSITION 2.2.4(a) For any £-presheaf F , L(F) is an £-mono-presheaf.(b) For any £-mono-presheaf F , L{F) is an £-sheaf.



Proof. — L(F) is an ^-mono-presheaf (resp. an <f-sheaf) if and only if hp o L{F) isan abelian mono-presheaf (resp. an abelian sheaf) for any tiny projective object P of£. Since it is clear that

hp o L(F) = L(hp o F)we are reduced to the case where £ = Ah which is well-known. D

DEFINITION 2.2.5. — We define the associated sheaf functor

A:Psh{X',£) -,Shv(X'^)by setting A = L o L. We have a canonical morphism

a(F) : F -,A(F).

PROPOSITION 2.2.6. — For any morphism

u : F ->Gfrom the presheaf F to the sheaf G there is a unique morphism

v : A(F) -^ G

making the diagrama(F)

F —4 A(F)

commutative. Therefore, F ^ A(F) if and only if F is a sheaf, and we have theadjunction isomorphism

K^^W^^0) ^ ̂ ^/.(W^G)which shows that Shv(X;£) is a reflective subcategory ofPsh(X',£).

Moreover, for any x C X , we have a canonical isomorphism

A(F).^F,.

Proof. — We haveh[A(F)][P]=A[h(F)[P]}.

Hence, there is a unique morphism

v'[P] : A[h(F)[P}} -^ h(G)[P]such that

vf[P]oa(h(F)[P})=h(u)[P].Since h is full there is a morphism

v : A(F) -> G

such thath(v) = v'.



For such a v^ we get

/i(v o a(F))[P] = v1 o a(h(F)[P}) = h(u)[P}

and since h is faithful, we have

v o a(F) = u.

Moreover, any morphism

w : A(F) -^ G

such that

w o a(F) = u

satisfies the equalityh{w) = v1

and h being faithful, we getw = v.

Since

h[A(F)}[P]=A(h{F)[P]),we get

hp(A(F),) = h[A(F)][P], ̂ h(F)[P], = hp(F,)

and the last part of the result follows easily. D

2.2.2. The category of sheaves

PROPOSITION 2.2.7. — The category

Shv(X^)

is quasi-abelian. Moreover, a sequence

E -> F -^G

is strictly exact (resp. coexact) in Shv(X;£) if and only if the sequence

E^F^-, G^

is strictly exact (resp. coexact) in £ for any x G X . In particular, a morphism

u: E -^ F

of Shv(X',£) is strict if and only if

u^ : E^ -, F^

is strict for any x € X .

SOClfiTfi MATHEMATIQUE DE FRANCE 1999


Proof. — Letu: E —^F

be a morphism in Shv{X\ <?). Define the object K of Psh(X, £) by setting

^(£/)=Kern([/)

for any open subset U of X. Since

h(K)=Kerh(u),

it is clear that K is an <?-sheaf. By construction, K is a kernel of u in Shv(X\£).Define the object C of P^(X; <?) by setting

C((7)=CokerH([7).

Since C is a cokernel of u in Psh(X\ <?), the adjunction formula for A shows that A(C)is a cokernel of u in <?^(X; <?).

It follows from what precedes that any morphism

u : E -^ F

of Shv(X'^ £) has a kernel, a cokernel, an image and a coimage and that

(Kern)a; ^ Kerâ.), (Im^ ^ Im(^),

(Cokern)a; ^ Coker(na.), (Coimn)^ ^ Coim^).Therefore, to conclude, it is sufficient to prove that u is an isomorphism if u^ is anisomorphism for every x G X. Since

h{u)[P}^hp{u,)

this is a direct consequence of the corresponding result for abelian sheaves. D

PROPOSITION 2.2.8. — The category Shv(X,£) is complete and cocomplete. More-over, direct sums and filtering inductive limits are strongly exact.

Proof. — LetF : J - ^ S h v { X ^ )

be a functor and let L be the projective limit of F in Psh(X,£). Since

hx o L = Hm hx ° F(j)j^J

for any object X o f < ? , L is an f-sheaf. Hence, L is a projective limit ofFin Shv{X\ <?).Let R be the inductive limit of F in Psh(X, <?). It follows from the adjunction formulafor A that A(R) is an inductive limit of F in Shv{X\£). The last part follows fromthe fact that in Shv(X',£) we have

(lunF(j)),=limFy),.3^J j e j

D



DEFINITION 2.2.9. — Let E be an object of E and let U be an open subset of X.Consider the <f-presheaf F defined by setting

W^ if vcu

[0 if V (/LUthe restriction map

r^y : F(V) -, F(W)being id^ if IV c V and 0 if W ^ V. By construction, for any <?-presheaf G, we have

^^rshW^G) ̂ Hom^G(^)).We set

Eu=A(F).Clearly,

K^^W^0) ^ Hom^G(£/)).

PROPOSITION 2.2.10. — For any open subset U ofX, the functor

(')u : £ -^Shv(X^)is strictly exact and preserves inductive limits. Moreover

fE if xeU(Eu)x ^ S\0 if xÛ

PROPOSITION 2.2.11. — Let Q be a strictly generating small set of objects of <?.Then,

[GU : G C Q, U open subset of X}is a strictly generating small set of objects of Shv(X'^£).

Proof. — Consider the canonical morphism

u : ^ F(U}u -^ Fue0p(x)

corresponding to the morphismF{U)u -^ F

deduced from the identity morphism

F(U) -> F(U).For any x € X, we have

( ^ F(U)u)^Q)F(U).uÔp(x) xeu

The morphism^F(£/)^F,U9x



induced by u^ corresponds to the restriction morphisms

r^:F(£/)^F,.

Hence, Uy: is a strict epimorphism for any x G X. This shows that u is a strictepimorphism. Since Q is a small strictly generating set of <?, for any U G Op(X),there is a small family (Gu,i)iîu °^ G an(^ a ^ri^ epimorphism of £

(9 Gu,i -^ F{U).ieiu

Since (')u preserves inductive limits,

Q){Gu,i)u-^F{U)uiElu

is a strict epimorphism in Shv(X',S), Hence, we get a strict epimorphism

(B 9(c;^)y^F.ue0p{x) ieiu


PROPOSITION 2.2.12. — The canonical inclusion

I : £ -> Cn(£)

gives a canonical functor

Shv(X;£) ->Shv(X^C/H(£)).

This functor induces an equivalence of categories

CH{Shv(X^)) ^Shv(X',Cn{£)).

Proof, — Since I is continuous, I o F is an CU{£) -sheaf for any <?-sheafF. This givesus a canonical functor

J : S h v ( X ' , £ ) -,Shv{X',CH(£)).

Ones checks easily that J is fully faithful and that its essential image is stable bysubobjects. Moreover,

J(F),^J(F,).

Let Q be a strictly generating small set of objects of <?. We know that I(Q) is agenerating small set of CT-L{£). Hence, for any object F of Shv{X\ CH{£)), there is asmall family (Ui,Gi)i^L of Op(X) x Q and a strict epimorphism

(3)(J(^))^F.I<EL

Moreover, since I commutes with filtering inductive limits, one checks easily that

{I(Gi))u, = J((Gi)u,)



and that9j((Go^)^j(^(Q)^).I^L ieL

Hence, we have an epimorphism

^((B(^)-^.l^L

The conclusion follows from Proposition 1.2.36. D

2.2.3. Internal operations on sheaves. — Let £ be a closed elementary quasi-abelian category and let T (resp. H, U) be its internal tensor product (resp. itsinternal homomorphism functor, its unit object).

DEFINITION 2.2.13. — Let F and G be two objects of Psh(X^). We denote byH(F, G) the kernel of the morphism

h: IJ H(F(U), G(U)) -^ U H(F(U)^G(V))uôp(x) uyÔpW

vcuof £ defined by setting

pny oh= H(\^F{u}-,^jj) opu - H(r^v,idG(v)) OPv

for any V C U in Op(X). Clearly,

U^H(F\u^G\u)

is a presheaf. We denote it by U(F,G\

PROPOSITION 2.2.14. — Assume F , G are two objects ofShv(X',£). Then, ̂ (F.G)is an object ofShv(X'^S).

Proof. — Let V C U be open subsets of X and let W be a covering of X. Since G isa sheaf, we have the strictly exact sequence

O->G(V)^ {J G(VF}W)^ J] G(vnwnw1).w^w w^w'ew

From the adjunction formula linking T and H , it follows that H ( F ( U ) , ' ) is a contin-uous functor. Therefore, we get the strictly exact sequence

0-^H(F(U),G(V))^ Y[ H(F(U),G(VnW))-^ JJ H(F(U),G(V H W n W ' ) ) .W(E\V W,W'^~W

Using a tedious but easy computation, we deduce from this fact that the sequence

0^^(F,G)(X)^ JJ U{F,G){W)-, J] U{F,G){W^W)M/ew W,W'EV^

is strictly exact in <f. And the conclusion follows. D



DEFINITION 2.2.15. — Let E, F be two objects of Shv(X,£). We denote T{E^F)the S -sheaf associated to the £-presheaf

U^T{E(UYF(U)Y

PROPOSITION 2.2.16. — We have the canonical functorial isomorphism

^^Sk^xÛE.F^G) ̂ Hom^^(^(F,G))forE, F, G inShv(X^).

Proof. — Let h G Horn shv{x^T(E^F^G) and let t/ D V be open subsets of X.By composition with the canonical morphism

T(E(U),F(V)) -. T(E(V)^F(V)) -^ T(^F)(Y),the morphism

h(V):r(E^F)(V)-^G(V)induces a morphism

huy:T(E(U)^F(V})-^G(y).By adjunction, this gives us a morphism

hfuy:E(U)^H(F(V)^G(V))

for any V C U in Op{X). Hence, we get a morphism

h'u :E(U) -^ n H(F{V)^G{V))vcu

V open

and one checks easily that h'{j factors through

h^:E(U)-^H{F\u^G\uYMoreover, the family (h'^)ue0p{x) defines a morphism

y(h) :E-^n{F,G)

mShv(X^).Now, let h G Horn ^^^.^.^ (£',%( jF',G)) and let U be an open subset of X. The

morphismh(U):E(U)^H{F\u^G\u)

gives rise to a morphism

h'(U):E(U)-^H(F(U^G(U)).

By adjunction, we get a morphism

h'Û) : T(E(U),F(U)) -^ G(U)

which is easily checked to be a morphism of presheaves. We denote

^ ( h ) : T { E ^ F ) ^ G


2.3. SHEAVES WITH VALUES IN AN ELEMENTARY ABELIAN CATEGORY 99

the associated morphism of <%v(X;<?). An easy computation shows that ip is aninverse of (p and that ^ and (p are functorial in £', F and G. D

COROLLARY 2.2.17. — The category Shv(X',£) endowed with T as internal tensorproduct, Ux as unit andU as internal homomorphism functor is a closed quasi-abeliancategory.

2.3. Sheaves with values in an elementary abelian category

2.3.1. Poincare-Verdier duality. — Let / : X —^ Y be a continuous map oflocally compact topological spaces and let A be an elementary abelian category. Recallthat

Shv(XÂ) (resp. Shv(Y',A))denotes the category of sheaves on X (resp. Y) with values in A. For short, we set

P*(X,A) = V^Shv{X,A)) (* = +,-,0)and use similar conventions for /C.

As usual, for any closed subspace Q of X and any A-sheaf F on X , FQ^X'.F)denotes the kernel of the restriction morphism

F(X)^F(X\Q).

DEFINITION 2.3.1. — For any sheaf F € Shv(X',A), we define the sheaf

MF)eShv(YÂ)by the formula

r([/;/.(F))= inn w1^);^)QCf-nU)^ /-proper

We call f-soft a sheaf F such that Fu is /-acyclic for any U e Op(X).

Of course, f\ is left exact and gives rise to a derived functor

a/.:^+(x;^)^p+(y;A).Hereafter, we will show that under the assumption that f\ has finite cohomological

dimension, Rf^ has a right adjoint functor. To get this result, we will adapt thereasoning of [7] to our more general situation.

DEFINITION 2.3.2. — Let K(F) denote a bounded functorial /-soft resolution ofF c Shv(X',A) (e.g. a truncated Godement resolution). For any G € Shv(Y',A), wedefine the presheaf

fK(G)^Psh(X'Â)through Proposition 2.2.2 by asking that

Horn (P, F([/; /^(G))) = Horn (f^K(Px)u)^ G)



for any P in a generating small set of small projective objects of A.

PROPOSITION 2.3.3. — For any G € Shv(Y\A), /^(G) G Shv(X,A). Moreover, ifG is injective, f^{G) is flabby.

Proof. — Let (Uz)iî be a covering of an open subset U of X. We know that, for anyk G Z, the complex

...^ ^K\Px)uû, -^(^K^Px)^ ^K\Px)u^îj^l iÎ

is exact. Since Kk(Px)v is /i-acyclic for any open subset V of X and /; has finitecohomological dimension, the complex

. . . ̂ (f) MK^Px^nu^) -^ 0 ̂ (Px)^) -^ f^K\Px)u) -^ 0i je l iÎ

is also exact. It follows that the sequence

0-^Rom(f^K(Px)u).G)-^Y[Hom(f,(K(Px)ui),G)-^ fl Hom(/!(X(Px)^n^-), G)id ijei

is exact. Hence, we see that the sequence0-^ Horn (P,r(l/;/^(G)))-^ Horn (P,nr(^,/^(G)))-^ Horn (P, Y[ Y(Ui n ̂ ; /^(G)))

iei ijeiis exact for any P in a small generating family of small projective objects. It followsthat the sequence

0^r(£/;/^(G))^nr(^;/i,(G))-^ ]̂ Y(UÛ^fK{G))iei ijei

is exact and that f^(G) is a sheaf. Let us assume now that G is injective. Let V bean open subset of U. We have the monomorphism

K ( P x ) v ^ K ( P x ) u .

Hence, we get the epimorphism

Horn (f,(K(Px)u).G)-^ Horn (MK(Px)v)^G).

As above, we deduce that

W/]<(G))^r(y;/i,(G))is an epimorphism. D

DEFINITION 2.3.4. — Let Z^y,*/!) denote the full subcategory of IC^~(Y,A) formedby complexes of injective sheaves. We denote by

f•:V+(Y;A)-^V+(XÂ)

the functor induced by/^^(y^^/c^x;^)



through the equivalence of categories

^(v^^p^y;^).PROPOSITION 2.3.5. — There is a canonical morphism of functors

Rf^-(G)-^G.

Proof. — Let G be an injective object of Shv(Y\ A). Since f^(G) is flabby, it is also/-soft and

Rf,f\G) ̂ f.f^G).By definition,

r(u;f,fK ((?))= inn W-û^f^G)}.QC/-1 ([/),<? /-proper

From the exact sequence

o ̂ roC/-1^);/^)) -> rv-1^); /],(<;)) ̂ n/-1^) \Q;/J^(G'))we deduce that Horn (P, ̂ (/"Û"); /]c(G'))) is a kernel of

Kom(f,(K(Px)f-nu)),G) -> Hom(f,(K(Px)f-nu)\Q),G)

Since the sequence

0 -^ K(Px)f-i(u)\Q -^ K(Px)f-^(u) -^ K(Px)q -^ 0

is exact, it follows that

Horn (P^QC-1^);^^)))^ Horn (/>(^(Px)o),G).

Since Q is /-proper, we have an obvious map

Pu -^ /.(PQ) -^ f^K(Px)Q).

Hence, there is a canonical morphism

Horn (P,rQ(/-1 (17);/^(G)))-^ Horn (P,r(£/;G))

which gives rise to a canonical morphism

ra/;/./i,((?))-^r(i/;G)as requested. D

THEOREM 2.3.6. — The canonical morphism

RHom (F, /'(G)) -^ RHom (A/.(F), G)

induced by the morphismRf,f\G)-^G

is an isomorphism.

SOClfiT^ MATHfiMATIQUE DE FRANCE 1999


Proof. — We know that any sheaf F G Shv{X^ A) has a resolution by sheaves of thetype

(BW)v.where V is an open subset of X and P a member of a generating small set of smallprojective objects of A. Since / has finite cohomological dimension, we may reduceourselves to the case where F = Pu and G is injective. In such a case, since f^(G) isflabby, we have

RHom(P^,/'(G)) ^Hom(P,r(£/,/^(G)))^Rom(f,{K(P)u).G)^RHom(P/,(Pt/),G)


2.3.2. Internal projection formula. — In this section, A denotes a closed ele-mentary abelian category with T as internal tensor product, U as unit object andH as internal homomorphism functor. We assume moreover that for any projectiveobject P of A, T(P, •) and H(P, •) are exact functors. It follows from the results inthe previous section that Shv(X\ A) endowed with T as internal tensor product, H asinternal homomorphism functor and U\ as unit object is a closed abelian category.

DEFINITION 2.3.7. — We say that an object P o!Shv(X',A) has projective fibers ifPa. is a projective object of A for any x G X.

LEMMA 2.3.8. — (a) Assume P is an A-sheaf with projective fibers. Then,

T(P, •) : Shv{X', A) -^ Shv(X^ A)

is exact. Moreover, if P' is another A-sheaf with projective fibers, then T{P^P') hasprojective fibers.

(b) Assume I is an injective A-sheaf. Then,

^(., I) : Shv{X', A)^ -^ Shv(X', A)

is exact. Moreover, if P is an A-sheaf with projective fibers, then ^(P,J) is aninjective A-sheaf.

Proof. — Part (a) follows directly from the fact that

T(^,F).=T(^,F,).

To prove the first part of (b), let

0 -> E' -^ E -> E" -^ 0

be an exact sequence of Shv{X\ A). Let P be a projective object of A and let U bean open subset of X. Since the ^4-sheaf Pu has projective fibers, it follows that thesequence

0 -^ T(P^, E ' ) -> T(Pu. E) -^ T(P^, E") -^ 0



is exact. Using the fact that I is injective, we get the exact sequence

0 -^ Horn (T(P^ E"), I ) -^ Horn (T(P^ E), I ) -^ Horn (T(P^ E'), J) -^ 0.

The adjunction formula between T and 7i then gives the exact sequence

0 -^ Horn {PuÊ", I ) ) -^ Horn (P^(E, I ) ) -^ Horn {Pu,H(E', I ) ) -^ 0.

Since for any A-sheaf F, Horn (P^/, F) ̂ Horn (P, F(£/)), we see that the sequence

0 -^ U(E',I)(U) -^ U{E,I)(U) -^ U(E",I)(U) -^ 0

is exact for any open subset U of X and the conclusion follows.The last part of (b) is obtained by similar methods. D

REMARK 2.3.9. — One can also prove that if I is an injective A-sheaf, then 1-L{E, I )is flabby for any .4-sheaf E.

PROPOSITION 2.3.10.— The functor

T : Shv{X', A) x Shv{X', A) -^ Shv(X^ A)

is explicitly left derivable and the functor

U: Shv(X', A)^ x Shv{X', A) -^ Shv(X-, A)

is explicitly right derivable. Moreover, we have the canonical functorial isomorphisms :(a) LT(E^F)^Lr(FÊ),(b) L T ( U x ^ E ) ^ E ,(c) RHom (LT(E, F),G) ̂ RHom (E, RU(F, G)),(d) RH(Ux,E)Ê.

Proof. — Let P denote the full subcategory of Shv(X', A) formed by A-sheaves withprojective fibers. By Proposition 2.2.11, any .4-sheaf F is a quotient of an object ofthe form

®W)y.t€J

where P, is a projective object of A and Ui is an open subset of X. Since

(^Wu} = (B P.,\iei ) y^ iei,Ui3x

it is clear that ©^j(Pz)^ belongs to P. Hence, any Asheafis a quotient of an objectof P. By the preceding lemma, it follows that (Shv(X;A),P) is T-projective. Hence,T is explicitly left derivable.

Denote Z the full subcategory of Shv(X',A) formed by injective A-sheaves. Weknow already that any object of Shv{X, A) is a subobject of an object of Z. By thepreceding lemma, (Shv(X\ A),T) is ^-injective. Hence, U is explicitly right derivable.The last part of the proposition follows directly by replacing the various objects bysuitable resolutions. Q



LEMMA 2.3.11. — Let X be a locally compact topological space. Let P be a protectiveobject of A and let E be an A-sheaf on X . Then, the canonical morphism

r(rc(x; E), P) -^ r,(x; T(E, ?x))is an isomorphism. In particular, T{E^Px) is c-soft if E is c-soft.

Proof. — We will work as in [7, Lemma 2.5.12].Without losing any generality, we may assume X is compact. Let (K^î be a

finite covering of X by compact subsets. Since E is an A-sheaf, we have an exactsequence of the form

o-^r{xÊ)^(^r(KÊ)^ (])r(^n^;E).id i je i

The object P being projective in A, the functor T(-,P) is exact and we get themorphism of exact sequences

0———>T(^(XÊ) ,P )— > ^Q)T(^(KÊ) ,P )—^ Q ) T ( r ( K i n K , ; E ) , P )iei i j e i

1^ h, ^ •4"

0 — — — > W r ( E , P x ) ) — — ^ Q ) W , T ( E , P x ) ) — — — ^ Q ) r ( K i n K , ; T ( E , P x ) )A iÎ ^ i j € l

Let us show that a is a monomorphism. It is sufficient to show that for any smallprojective object Q of A and any

h:Q-^T(T(XÊ)^P)

such that a o h = 0 we have h = 0. Since

(*) Inn T(r(£7;E),P)^r(^P)^ Inn r((7;T(^Px))U3x U3x

U open U open

and Q is tiny (see Remark 2.1.2), we can find a finite compact covering of X suchthat

A o h = 0.

Since A is monomorphic, the conclusion follows.To show that a is epimorphic, it is sufficient to show that for any small projective

object Q of A and anyA:Q^r(X;T(E,Px))

there ish' :Q->T(r(X;E),P)

such that a o h1 = h. Using once more (*) and the fact that Q is tiny, we can find afinite compact covering of X such that

A' o h = 13 o h"



for some

^:Q_,^T(r(^;E),P).»€J

It follows from the first part of the proof that f3 and 7 are monomorphic. Since

0 = // o V o h = // o f3 o ̂ = 7 o ̂ o ///

we see that fi o h" = 0. Hence,

/z" = A o h'

for some

^:Q-^T(r(X;E),P).

For such an /i', we have

A' o a o h1 = f3 o X o h' = f3 o h'1 = V o h.

Hence, a o h' == h and the proof is complete. D

LEMMA 2.3.12. — Let f : X —^ Y be a morphism of locally compact topologicalspaces. Let P be an A-sheaf on Y with projective fibers and let E be an A-sheaf onX . Then, the canonical morphism

rcf^P)-^/!^/-1?)is an isomorphism. Moreover, T(E,f~lP) is f-soft if E is f-soft.

Proof. — Since

T{f,E,P\ = T((f,E)y,Py) = TW\y);E),Py)

and

f,(r(E,f-lP))y=W-l(y)•,T(E\f-^),(Py)\f-^y)))

for any y G V, we are reduced to the preceding lemma. D

PROPOSITION 2.3.13. — Let f : X —f Y be a morphism of locally compact topologicalspaces. Assume f\ has finite cohomological dimension. Then,

LrWÊ^F) ̂ RfWEJ^F)

for any E in V~ (Shv (X', A)) and any F in V~ {Shv (Y', A)).

Proof. — This follows directly from the preceding lemma if we replace E by a softresolution and F by a resolution by A-sheaves with projective fibers. D



2.3.3. Internal Poincare-Verdier duality

PROPOSITION 2.3.14. — Letf : X -^ Y be a morphism of locally compact topologicalspaces. Assume f\ has finite cohomological dimension. Then, the canonical morphism

Rf^Rn(E,fF) -, RH(Rf,E,F)induced by

Rf^fF -^ Fis an isomorphism in V^(Shv(YÂ)) for any E in P- (Shv (X; A)) and any F inP+(5^(V;A)).

Proof. — It is sufficient to prove that

Rrcr^R^/'F)) -^ Rr(y;R^(j?/,^F))is an isomorphism for any open subset V of Y. We may even restrict ourselves to thecase V = Y and prove only that

RHom (P, Rr(V; RH(E^ f-F))) ̂ RHom (P, RF(X; RH{Rf,E^ F)))

for any projective object P of A. This follows from the chain of isomorphism below:

RHom(P,Rr(y;R^(EJ'F))) ^ RHom(Py,R^(EJ'F))

^RHom(Lr(E,Py)J'F)

^ RHom (Rf,LT(E, f^Px), F)^RHom(LT(^/,^Px),F)^ RHom (Px, RH(Rf,E, F))^ RHom (P, Rr(X; Wi{Rf,E, F))).

D


CHAPTER 3

APPLICATIONS

3.1. Filtered Sheaves

3.1.1. The category of filtered abelian groups. — To fix the notations, let usrecall the following definitions.

DEFINITION 3.1.1. — A filtration on an abelian group M is the data of an increasingsequence (-Ffeêz of abelian subgroups of M such that

U Fk = M.feez

A filtered abelian group M is an abelian group Moo endowed with a filtration

(Mfe)fcez.

We call Moo the underlying abelian group of M.A morphism of filtered abelian groups u : M —^ N is the data of a morphism

Hoc : Moo -^ Aôo

of the underlying abelian groups such that

ôo(Mfc) C Nk

for any k (E Z. The set of morphisms from M to N is denoted

Hom(M,AO.

It is clearly endowed with a canonical structure of abelian groups. With this notion ofmorphisms, one checks easily that filtered abelian groups form an additive category.We will denote it by JAb.

The following two obvious propositions will clarify the structure of limits in TAh.

PROPOSITION 3.1.2. — The category JAh has kernels and cokernels. More precisely,let u : M —^ N be a morphism of filtered abelian groups. Then,

108 CHAPTER 3. APPLICATIONS

(a) Kern is the abelian group u^^Q) endowed with the filtration

(u^(O)nMk)^

(b) Cokern is the abelian group Aô/ôo(Xoo) endowed with the filtration

(Nk + ̂ oo(Moo)/Hoo(Moo))fcez-

A s a consequence, we see that(c) Im u is the abelian group Hoc (MX)) endowed with the filtration

(u^{M^)r}Nk)kez.

(d) Coimu is the abelian group Moo/u^(0) endowed with the filtration

(Mk+u^W/u^(0))^

It may equivalently be described as the group Uoo(Moo) endowed with the filtration

(uoo(Mk))kez-

In particular, the morphism u is strict if and only if

u^(Mk) = Uoo{M^) H Nk

for every k G Z.

PROPOSITION 3.1.3. — The category TAh has direct sums and products. More pre-cisely, let (Mi)iî be a small family of filtered abelian groups. Then,

(a) ®^j Mi is the abelian group ®^j(^)oo endowed with the filtration

f®(M^) ,Y^ ) k^L

(b) n%e^ ̂ i ^s ê a^ân subgroup

U IP^k<EZi€l

ofY[iî(Mi)oo endowed with the filtration

frr^) •vez / feez

REMARK 3.1.4. — It follows from the last point of Proposition 3.1.2 that FAb is notabelian. Moreover, Proposition 3.1.3 shows that ifJ is infinite, (]~[^ ^)oo m8iy d^61*from n^W)00-

PROPOSITION 3.1.5. — The category TAh is a complete and cocomplete quasi-abeliancategory in which direct sums and filtering inductive limits (resp. products) are stronglyexact (resp. exact).

Proof. — It is direct consequence of the two preceding propositions. D


3.1. FILTERED SHEAVES 109

DEFINITION 3.1.6. — Let M be a filtered abelian group and let I G Z. We denoteby M(l) the filtered abelian group obtained by endowing Moo with the filtration

(Mi^k)kez'

Clearly,M ̂ M(l)

is a functor of TAb into itself. We call it the filtration shifting functor. Let M, N betwo filtered abelian groups.

The sequence

(Hom(M,7V(fe)))feez

of subgroups ofHorn (M, N)

is increasing and gives a filtration of

|j Hom(M,A^)).A;ez

We denote FHom (M, N) the corresponding filtered abelian group.Denote (M 0 N)k the image of the canonical morphism

Q) Mi (g) A^ -^ Moo 0 N^ôo y^ -^00l^L

induced by the canonical inclusions

Mi -^ Moo, Nk-i -^ Aôo.

Clearly, ((M 0 N)k)k€Z forms a filtration of Moo ^ Aôo- We denote by M 0 N thecorresponding filtered abelian group.

Finally, we denote by FZ the filtered abelian group obtained by endowing Z withthe filtration defined by setting

FZ.={ 2 "^|^0 otherwise.

PROPOSITION 3.1.7. — The category TAh endowed with -0- as internal tensor prod-uct, FHom^,-) as internal Horn-functor and FZ as internal unit forms a closedadditive category. In particular, we have

(a) Horn (M (g) N, P) ̂ Horn (M, FHom (N, P)),(b) M(g)7V^7V(g)M,(c) M(g)FZ^M,

for any objects M, N , P of JÂb.



PROPOSITION 3.1.8

(a) For any I € Z, we have

FHom(FZ(-0,M)^Mz

for any object M of JAb. In particular, FZ(—^) is a tiny projective object of JAb.(b) For any object M of JAb, the canonical morphism

(D (])FZ(-Q-^Ml^L h€Mi

induced by the preceding isomorphism is a strict epimorphism. In particular,

(FZ(-O)^z

forms a strictly generating family of objects of JAb.

COROLLARY 3.1.9. — The category JAb is an elementary closed quasi-abelian cate-gory. In particular, JAb has enough projective objects. Moreover, for any projectiveobject P of JAb, the functor

P 0 < : JAb -^ JAb

is strongly exact and P 0 P' is projective if P' is projective.

To show that JAb has enough injective objects, we need first a few auxiliary results.

DEFINITION 3.1.10. — For any filtered abelian group M, we denote by M(oo) thefiltered abelian group obtained by endowing Moo with the constant filtration.

PROPOSITION 3.1 .11. — Assume R is a cogenerator of Ab. LetFR denote the objectof JAb obtained by endowing the abelian group R with the filtration defined by setting

_ { R ifk>0,r Hk — \

[0 otherwise.

Then, n FR^Jfe€ZU{oo}

is a strict cogenerator of JAb.

Proof. — Let M be an arbitrary filtered abelian group. We have to show that thecanonical morphismz : M ^ n ( n FR^

hCHom^M^^^FRW) ^€ZU{oo}

is a strict monomorphism. First, note that

Hom^(M, JJ FR(k))= ^ Hom^(M,FJ?(A-))

and that

A;GZU{oo} fcGZU{oo}

Rom^(M,FR(k))


3.1. FILTERED SHEAVES HI

is the subset of Horn (Moo, -R) formed by morphisms hk : M —^ R such that

hk(M-k-i) =0,where we set M-oo = 0 by convention. Therefore, to give

he}iom^(M^FR(k))kez

is to give a family(hk : Moo -^ R)kez

of morphisms ofabelian groups such that hk(M-k-i) = 0. Moreover, for any m G M,we have

[z(rn)h]k = hk{m)for any h C Hom^(M, fêz ̂ W) and any ^ e Z.

Let us show that i is a monomorphism. Assume m G Moo \ {0}. Since R is acogenerator of Ab, we can find a morphism

hoo '• Moo —)- -Rsuch that hoo(m) / 0. Setting hj, = 0 for any k € Z, we get a morphism

h E Hom^(M, JJ FA(A-))feez

such that[z(rn)h]oc + 0.

Hence, z(m) 7^ 0 and the conclusion follows.Let us now prove that i is strict. Assume m e Moo is such that i(m) has degree

less than L This means that[z(rn)h]k e FRk+i

for any h G nHom^^(M, rLezôo}^^)) and any A- C Z. Therefore,

hk(m) = 0

for any k < -1. Assume m ^ Mi. Denote p : Moo —> Moo/M/ the canonicalprojection. Since

p(m) / 0,

we can find a morphism h1 : M^/MI —^ R such that h'(m) ^ 0. Consider themorphism

h G Hom^(M, JJ FR(k))

defined by settingA;GZU{oo}

( h ' o p [fk=-l-lhk = \

10 otherwise.We get a contradiction sincee

/^-i(m) / O



and -I - 1 < -/. Therefore, m C Mi and the proof is complete. D

PROPOSITION 3.1.12. — Assume I is an injective object of Ab. Then,

FI(k)

is an injective object of FAh for any k C Z U {00}.

Proof. — Let M be an object of FAb. For k = oo, we have

Hom^(M,FJ(oo)) = Hom^(M^J)

and the result is obvious. Let us assume k ^ oo. In this case, we have

Hom^(M,FJ(A-)) = Hom^(Moo/M_,-i,J).

Let

0 -^ M' -^ M -> M" -^ 0

be a strictly exact sequence in FAb. We get a commutative diagram of Ah

0 0 0

1 1 10 ————> M^_i —————> M_fe_i —————> M"k-i ————> 0

I 1 10 —————> M'^ ———————> Moo ———————> M^ —————> 0

[ 1 1M^/M'_,_, ——> MÎM.k-v ——> M^/M'^-z

[ [ [0 0 0

where all the columns and the first two lines are exact. Therefore, the last line is alsoexact. Since I is injective in Ah^ the sequence

0 ̂ Hom(M^/M^,_i,J) ̂ Horn (Moo/M-,_i,J) ̂ Hom(M^/M^_i,J) ̂ 0

is exact. This shows that F I ( k ) is injective in FAb. D

COROLLARY 3.1.13. — The category FAb has enough injective objects.

Proof. — Apply the preceding propositions to an injective cogenerator of Ab (e.g.Q/^). D



3.1.2. Separated filtered abelian groups. — Let us recall that any filteredabelian group M may be turned canonically into a topological abelian group by taking[Mk : k C Z} as a fundamental system of neighborhoods of 0 in Moo. In the sequel,when we apply topological vocabulary to a filtered abelian group, we always have thisparticular topological structure in mind. In particular, a filtered abelian group M isseparated if and only if f^cz ̂ = 0.

DEFINITION 3.1.14. — We denote FAb the full subcategory of FAb formed by sepa-rated filtered abelian groups.

PROPOSITION 3.1.15. — Let (Mi)î be a family of separated filtered abelian groups.Then, the filtered abelian groups

@Mi and {JM,id iel

are separated. In particular, they form the direct sum and direct product of the family(Mi)î in FAb.

PROPOSITION 3.1.16. — The category FAb has kernels and cokernels. More pre-cisely, let u: M —^ N be a morphism ofFAb. Then,

(a) Ker u is the subgroup u^ (0) endowed with the filtration

(u^\0)nMk)ke^

(b) Cokern is the group N^/u^(M) endowed with the filtration

(A^+ôo(M)/ô(M))^z.

Hence,(c) 1m u is the group Uoo(M) endowed with the filtration

(u^{M)F}Nk)ke^

(d) Coimu is the group Moo/u^(0) endowed with the filtration

(Mk+u^(0)/u^(0))^

It is isomorphic to the group Uoo(Moo) endowed with the filtration

(u^{Mk))kez-

In particular, u is strict in FAb if and only if it is strict in FAb and has a closedrange.

PROPOSITION 3.1.17. — The category TAb is quasi-abelian.

SOCIETfi MATHfiMATIQUE DE FRANCE 1999


Proof. — We know that FAb is quasi-abelian. By the characterization of the strictepimorphisms in JAh and the structure of kernels in TAh^ the axiom (QA) is auto-matically satisfied. Let us deal with the axiom (QA*). Consider the co-cartesiansquare

M' -^N'

v v'

M-^N

of JÂb. Assume u is a strict monomorphism of FAb. This means that u is a strictmonomorphism of TAh and that its range is closed. We know that

w : M —^ N C M'm ^-> (u(m)^v(m))

is a strict monomorphism of JAh. Let {mk)ke^ be a sequence of M such that

w(nk) —^ (n,m')

in N C M ' , it follows thatu(mk) —>• n

in N. Since u is a strict monomorphism with closed range, there is m G M such that

mk —^ rain M and u(m) = n. Therefore,

v(mk) —f v(m)

and since M' is separated, we get

v(m) = m'.

Hence,w(m) = (n.m')

and we see that the range of w is closed. Therefore, the sequence

O - ^ M - ^ N e M ' - ^ N ' - ^ Q

is strictly exact in TAh. It follows that u' is a strict monomorphism of FAb and that

Cokern ^ Cokerz/.

Since Cokern is separated, Coker?/ is also separated and u1 has a closed range. D

PROPOSITION 3.1.18. — In JAh,

(FZ(Q)^zforms a small strictly generating family of small protective objects. In particular, JAbis a quasi-elementary quasi-abelian category.



Proof. — This follows directly from Proposition 3.1.15 and Proposition 3.1.16 sinceFZ is clearly separated. D

REMARK 3J-.19. — The object FZ is not tiny since there are filtering inductivesystem of TAh with a non zero inductive limiUn TAh but a zero inductive limit inFAb. As an example, consider an object M of JAh and the system

(M(0),ez.

In TAh^ we see easily thatInnM(Z)/ez

is the group Moo with the constant filtration. Hence, in JAh,

limM(0 ^0.l^L

DEFINITION 3.1.20. — We denote by

J : FAb -^ FAb

the canonical inclusion functor. We denote by

L : FAb -^ FAb

the functor defined by£(M) =M/^kezMk.

PROPOSITION 3.1.21. — There is a canonical adjunction isomorphism

Hom^(M,?(AO) ^ Hom^(L(M),AO.

In particular, the functor I is compatible with projective limits and the functor L iscompatible with inductive limits.

PROPOSITION 3.1.22. — The functor

J : ^FAb -^ JAb

is strictly exact and induces an equivalence of categories

?:P(^4&) -^VÂb).

Its quasi-inverse is given by

LL:T){yAb) -^V(Âb).

Through this equivalence, the left t-structure of P(^4&) is exchanged with the leftt-structure of V^FAb). In particular,

? -.CnÂ^wCHÂb).

SOClfiTtS MATHEMATIQUE DE FRANCE 1999


Proof. — Since any projective object of FAb is a direct factor of a filtered abeliangroup of the form

(DFZ(^)i€l

which is separated, it is also separated. Let M' be an object of D(JAb) and let

P- -4 J(M-)

be a projective resolution of J(M'). Since the components of P' are separated,

£(?•) ^ P- ^ M-

in D(^4&). Hence,

LL o 7 ̂ id .

Let

P ^M-

be a projective resolution of M' C D{FAb). Since the components of P ' are separated,we have

L(P-) ^ P-

in D(7Ab) and

J o £ ( P ' ) ^ M -

in D{FAb). Hence,

7 o LL ^ id.

To conclude, it is sufficient to remark that a sequence

M' -, M -^ M"

of FAb is strictly exact in FAb if and only if it is strictly exact in FAb. D

REMARK 3.1.23. — The functor

J : v(FAb) -^ V{FAb)

does not preserve the right t-structures. As a matter of fact, if u : M —^ N is a strictmonomorphism of FAb with a non closed range, the complex

0^ M -^N -^0

with M in degree 0 has null cohomology in that degree in KT-L^FAb) but not inmi(FAb).



3.1.3. The category 7^ and filtered sheaves. — Hereafter, we will have to dealwith graded rings. Our point of view is that these are rings in the closed abeliancategory of graded abelian groups and graded additive maps. We denote the internaltensor product of this category by 0 and its internal Horn functor by GHom. Con-sequently, by a module over a graded ring we always mean a graded module and bya morphism between such modules will always mean a graded morphism.

DEFINITION 3.1.24. — Let RZ denote the graded ring Z[T].To any filtered abelian group M, we associate the graded RZ-module

R{M)=Q)M^feez

the multiplication•T : Mk -^ Mfc+i

being the canonical inclusion. This gives us an additive functor

R: FAb -^ Mod{W)

where Mod(R^) denotes the category of RZ-modules.Let N be an RZ-module. Denote

nfe+i^ : Nk -> Nk-^-i

the action of T and consider the inductive system

(TV^n^+i^êz.

Set

(L(AO)oo=lm^feez

and let (L(N))k be the canonical image of Nk in (L(A^))oo« Clearly,

(L(AQ,),ez

forms a filtration of the abelian group (L{N))oo. We denote L{N) the correspondingfiltered abelian group. This gives us an additive functor

L : .Mod(RZ) -^ FAb.

PROPOSITION 3.1.25. — We have the canonical functorial isomorphisms

Hom^(L(AO,M) ̂ Hom^^(7V,J?(M))

and

L o R(M) ̂ M.

In particular, R is a fully faithful continuous functor and L is a cocontinuous functor.Moreover, R is strictly exact and is compatible with direct sums.

SOCltSTfi MATHfiMATIQUE DE FRANCE 1999


PROPOSITION 3.1.26. — The essential image of

R: JAb -^ A-W(RZ)

is formed by RZ -modules M such that

T' : M -> M

is injective.

PROPOSITION 3.1.27. — The essential image of R forms a L-projective subcategoryof Aôc^RZ). In particular,

L : Aôd(RZ) -^ JAb

is an explicitly left derivable right exact functor which has finite homological dimen-sion.

Proof. — Let us denote P the essential image of R.(a) Any object of Aôd(RZ) is a quotient of an object of P. As a matter of fact,

the canonical morphism^ (f) RZ(-fe) ̂ Mke^hCMk

is an epimorphism for any RZ-module M and it follows from the preceding proposi-tions that

Q) Q) RZ(-k)^R((^ (3) FZ(-fc)).kezheMk kezheMk

(b) In an exact sequence

0 -> M' -^ M -^ M" -^ 0

of Mod{W} where M, M11 belong to V, M belongs to P. This follows directly fromthe preceding proposition since a subobject of an object of P is clearly an object ofP.

(c)If0 -^ M' -^ M -^ M" — 0

is an exact sequence of Alod(RZ) with M ' , M, M" in P, the sequence

0 -^ L{M') -^ L(M) -^ L(M") -^ 0

is strictly exact in TAh. As a matter of fact, we may assume

M' ^ R(N'), M ̂ R{N), M" ^ R{N").

Hence, using the fact that R is fully faithful, we see that the given exact sequencemay be obtained by applying R to a strictly exact sequence of the form

0 -^ N ' -> N -^ N11 -^ 0

of FAb. The conclusion follows from the fact that L o R ̂ id. D




RR : V{FAb) -> P(.Mod(RZ))

is an equivalence of categories which exchanges the left t-structure ofV{JAb) with thecanonical t-structure o/T>(.Mod(RZ)). A quasi-inverse of RR is given by

LL : V(Mod{W)) -> V{JAb).

In particular, R induces an equivalence of categories

CH(JAb) w Mod(W).

Proof, — This follows directly from the preceding propositions. D

COROLLARY 3.1.29. — The functor

JAb -^ Ab^

which associates to any filtered abelian group M the inductive system

k ̂ Mfe,

the transitions of which are given by the inclusions

Mk -^ Mk' {k ^ A/),

induces an equivalence of categories

Cn(JAb) w Ab^.

Proof. — This follows from the preceding proposition if one notes that the functor

Mod(W) -^ Ab^which sends an RZ-module M to the inductive system

(M^T.:M^Mfc/)^zis clearly an equivalence of categories. D

PROPOSITION 3.1.30. — The structure of closed category of JAb induces a structureof closed category on CT-L(JAb) which is compatible with the usual structure of closedcategory of .Mod(RZ) through the equivalence of the preceding proposition.

Proof. — Note that there are obvious canonical functorial morphisms

R(M) 0 R(N) -^ R(M 0 N)R(¥Rom{M,N)) -, GHom (fl(M), R(N))

for M, N in FAb. Although they are not bijective in general, one can check easilythat they become isomorphisms if M ̂ FZ(() for some / € Z. Therefore, we see that

RR(M) (^L RR(N) ̂ RR(M (g^ N)

.RJ?(RFHom(M,AO) ^ RGRom{RR{M), RR(N))and the conclusion follows. D



REMARK 3.1.31. — The category K = CHÂb) is a closed elementary abelian cat-egory for which we may apply all the results obtained in the preceding chapter. Inparticular, the cohomological properties of filtered sheaves on a topological space Xare best understood by working in P(X,7^).

3.2. Topological Sheaves

3.2.1. The category of semi-normed spaces. — In this section all vector spacesare C-vector spaces. Recall that a semi-norm on a vector space E is a positive functionp on E such that

p(ei+62) <p(ei)+p(e2) for any e^.e^Êp(ce) = \c\p(e) for any c G C, e G E

Let E be a vector subspace of F and let p be a semi-norm on F. Recall that pinduces a semi-norm p ' on E and a semi-norm p" on F / E . These semi-norms aredefined respectively by

p\e)=p(e) and p"([/]^) = jnp(/ + e).

DEFINITION 3.2.1. — A semi-normed space^ is a vector space endowed with a semi-norm RE- A morphism of semi-normed spaces is a morphism / : E —> F of theunderlying vector spaces such that

\PF o /| < ORE

for some C > 0. With this notion of morphisms, semi-normed spaces form a categorywhich we denote by Sns.

Let E be a semi-normed space. As is well-known, the semi-norm RE gives riseto a canonical locally convex topology on E. Hereafter, we will always have thisparticular topology in mind when we use topological vocabulary in relation withsemi-normed spaces. Using this convention, a morphism of semi-normed spaces issimply a continuous linear map.

LEMMA 3.2.2. — The category Sns is additive. More precisely:

(a) The C-vector space 0 endowed with the 0 semi-norm is a null object of Sns.(b) For any E,F in Sns, the C-vector space E © F endowed with the semi-norm p

defined byP(e,f)=pE(e)+pF(f)

is a biproduct of E and F in Sns.

LEMMA 3.2.3. — Let f : E —)- F be a morphism of Sns. Then,(a) Ker/ is the vector space f^W endowed with the semi-norm induced by RE;(b) Coker/ is the vector space F/ f(E) endowed with semi-norm induced by pp.


3.2. TOPOLOGICAL SHEAVES 121

Therefore,

(c) Im/ is the vector space f{E) endowed with the semi-norm induced by pp;(d) Coim/ is the vector space E/f^Ô) endowed with the semi-norm induced by

PE- Equivalently, Coimf may be described as the vector space f(E) endowedwith the semi-norm p defined by

p(x) = mf pE(y)yef-1^)for any x G f(E).

In particular,

(e) / is strict if and only if

mf pE{x + y ) < Cpp(f(x))

for some C > 0. In other words, f is strict if and only if it is relatively open.

PROPOSITION 3.2.4. — The category Sns is quasi-abelian.

Proof. — We know that Sns is additive and that any morphism of Sns has a kerneland a cokernel.

Consider the cartesian square

E^FT T.1 |,T-^G

where / is a strict epimorphism and let us show that u is a strict epimorphism. Wemay assume that T is the kernel of

E^G^-^F.Hence,

T = { ( x ^ y ) ^ E e G : f { x ) = g ( y ) } .Denoting

i : Ker (/ -g) -^ E C Gthe canonical injection and TTE and TTQ the canonical projections, we have

v = 7rg o i and u = TTG ° i '

Since / is surjective, for any y G G there is x € E such that

fW=g{y).In such a case, (x,y) € T and

u(x,y) =y.This shows that the application u is surjective.



Recall that

PE@G((x, y)) = pE(x) + pG(y) ^(x, y ) G E C G.Clearly,

Kern = {(e, 0) C E C G : /(e) = 0}.Therefore, for any (a;, y ) c T, we have

^K^^ + (e?e^ = ee^/^ + e) +^'

Since / is a strict epimorphism, there is C > 0 such that

^lfPE{x+e)<CpF{f(x))

for any x <E E. From the continuity of g , we get C ' > 0 such that

P F ( f ( x ) ) = p F ( g ( y ) ) ^ C t p G ( y ) .for any (a;, 2/) G T. Hence, there is C11 > 0 such that

(e^L.^^^^^^^^^^

for any (a*. ?/) G T and u is a strict epimorphism.Consider the cocartesian square

G-a-^Tt T

"I I"E^F

where / is a strict monomorphism and let us show that u is a strict monomorphism.Denote a the morphism

(-^) : E ^ G e F .We may assume that

T = Cokera = (G C F ) / a ( E ) .Denoting

^GeF-^ ( G ( ^ F ) / a ( E )the canonical morphism and (TF and (TG the canonical embeddings, we have

u = q o (JQ and v = q o <jp.

Consider y C G such that n(^/) = q o cr^(^) = 0. It follows that

( y ^ ) C a ( E )and there is x C £' such that

{y^)=(g(x)^-f(x)).Since / is injective, x = 0 and we get y = g(x) = 0. Hence u is injective.



Since g is continuous and / is strict, we can find positive constants C, C ' and C"such that

po(y) <: pc(y + g(x)) + pc{g{x))< l p G ( y - ^ g ( x ) ) +CpE(x)< p G { y + g ( x ) ) + C ' p F ( f ( x ) )<C"pG^F({y+gW^-f(x)))

for all y G G and all x C E. Therefore, for any y G G, we have

PcQ/) ^ ̂ / i n f . p ( ( y + ̂ ), -/Or)))xÊ

<C" inf p(Q/+^))(^QeQ'^)

^%((^0)))

^ ^p(n^)),

where p denotes the semi-norm of {G 0 F ) / a ( E ) induced by p. It follows that u is astrict monomorphism. D

DEFINITION 3.2.5. — Let E and F be semi-normed spaces.We denote by E^F the semi-normed space obtained by endowing the vector space

E 0^ F with the semi-norm p defined by

p(z) = inf P E ( x k ) p F ( y k ) 'z=^k=l -^fc^fc

We denote by L (£', F) the vector space Horn ̂ (I?, F) endowed with the semi-normq defined by

q{h) = sup pF(h{x)).PE{X)<1

We denote by C the semi-normed space obtained by endowing C with the semi-norm | • [.

PROPOSITION 3.2.6. — The category Sns endowed with 0 as internal tensor product,L as internal homomorphisms functor and C as unit object form a closed category.

DEFINITION 3.2.7. — Let (Ei)iî be a family of semi-normed spaces. We denote by^ Ei the vector space (B^j Ei endowed with the semi-norm p defined byid

P((ei)î) = ̂ pi{ei).i€l

We denote by ]~[ E^ the subvector space of Y[î Ei formed by the families (e^)^jiei

such thatp(e) = suppi(ei) < +00

iÎ



endowed with the semi-norm p.

PROPOSITION 3.2.8. — (a) Let (ui : Ei —)• F)iî be a bounded family of morphismsof semi-normed spaces. Then, there is a unique morphism

u:(Êi-^Fiei

such that u o Si = Uz (here S i : Ei —^ Q Ei denotes the canonical monomorphism).iî

(b) Let (vi: F —^ Ei)iî be a bounded family of morphisms of semi-normed spaces.Then, there is a unique morphism

v:F-t\[E,iei

such that pi o v = Vi (here p i : ]~[ —^ Ei denotes the canonical epimorphism).iei

(c) Let (ui: Ei —^ Fi)iî be a bounded family of morphisms of semi-normed spaces.Then, the kernel and the cokernel of

(D^9îÎ iÎ

are respectively isomorphic to Q) Kerui and Q) Cokerui. Similarly, the kernel ofid i<El

n^nîÎ iCi

is isomorphic to Y[ Kerui. Moreover, if each Ui is strict, then the cokernel ofiei

n^-nîei iei

is isomorphic to ]~[ Cokerui.iei

REMARK 3.2.9. — We could also introduce the subcategory Sns of Sns whose mor-phisms are the linear maps / : E —^ F such that

\PF ° f\< P E '

Then,

e and nid id



appear as the true direct sum and direct product of objects of Sns. Note however that

Sns is not additive and so does not enter into the general framework of quasi-abeliancategories.

COROLLARY 3.2.10. — For any set I

(^C ( r e s p . ^ [ C )iEl iEl

is projective (resp. injective) in Sns.

Proof. — The first part follows directly from the preceding proposition thanks tothe characterization of strict epimorphisms contained in Lemma 3.2.3. As for thesecond part, the characterization of strict monomorphisms (loc. cit.) reduces it to thewell-known Hahn-Banach theorem. D

PROPOSITION 3.2.11. — The category Sns has enough projective and injective ob-jects.

Proof. — (a) For any object E of Sns^ the canonical morphism

<^CÊbe BE

defined byu{{Cb)b€bE) = ̂ Cbb

bCBE

is a strict epimorphism. As a matter of fact, any b' e BE may be written as

u((8b'b)b€BE)and

P ^ ^b'b)b(EBE = L

b^BE

Thanks to the preceding corollary, it follows that Sns has enough projective object.(b) Let E be an object of Sns and let us show that there is a strict monomorphism

from E to an injective object of Sns. Denote N the subspace p~l(0) of E endowedwith the null semi-norm. Since any linear map h : X —^ N is continuous, it is clearthat N is injective in Sns. Therefore, the sequence

0 -4- N -> E -^ E / N -^ 0

splits in Sns and E is isomorphic to N (D E / N . Hence, we may assume N = 0 (i.e. Eis separated). In this case, denote by B^ the unit semi-ball of L(£J,C) and considerthe morphism V : E -> n c

<p€B^



defined byv(e)y =^(e).

Thanks to the theorem of bipolars, this is clearly a strict monomorphism and theconclusion follows from the preceding corollary. D

PROPOSITION 3.2.12. — Let (Ei)iî be a family of semi-normed spaces. Then, forany semi-normed space F we have the following canonical isomorphisms

(^(E,0F)^((Êi)^Fiel i<El

L((J)E^F)^f[L(E^F)id

L(FÊi)^f[LÊi)iÎ

Proof. — This follows directly from the adjunction formula

Hom^ (E^)F,G) ̂ Horn ̂ (E,L(F,G)).Sns Sns

D

PROPOSITION 3.2.13. — For any projective object P of Sns the functor

P 0 • : Sns —^ Sns

is strongly exact. Moreover for any projective object P' of Sns, the object P 0P' isalso projective.

Proof. — Let P be a projective object of Sns. Since the result will be true for a directfactor of P if it is true for P, we may assume that P is of the form ̂ C. Thanks to

i(ElPropositions 3.2.12 and 3.2.8 we may even reduce ourselves to the case P = C. ButC is the unit object of the closed category Sns^ so we get the conclusion. D

COROLLARY 3.2.14. — The abelian category C'H(Sns) has a canonical structure ofclosed category.

Proof. — This follows from Corollary 1.5.4. D

3.2.2. The category of normed spacesDEFINITION 3.2.15. — We denote by Afvs the full subcategory of Sns formed bynormed vector spaces.

PROPOSITION 3.2.16. — Let u: E —^ F be a morphism of Afvs. Then(a) Keru is the subspace u~l(0) endowed with the norm induced by that of E,(b) Cokern is the quotient space F/u(E) endowed with the norm induced by that of

F,



(c) Imu is the subspace u(E) endowed with the norm induced by that o f F ,(d) Coimu is the quotient space E/u-1^') endowed with the norm induced by that

ofE,(e) u is strict if and only if u is relatively open with closed range.

Proof. — Direct. Q

PROPOSITION 3.2.17. — The category At vs is quasi-abelian.

Proof. — Since Afvs is clearly additive with kernels and cokernels, we only need toprove that axiom QA is satisfied. Let

EO^F,

Ei-^F,

be a cartesian square with u a strict epimorphism. It follows from the precedingproposition that UQ is a strict epimorphism of Sns and that the square is cartesian inSns. Therefore, u\ is a strict epimorphism in Sns and thus in Afvs. Now, let

F ul ^ rcj\ ——> ̂ iT T

.J ] f

EO^F,

be a cocartesian square in Afvs where UQ is a strict monomorphism ofAfvs. It followsthat UQ is a strict monomorphism of Sns with closed range. By definition the sequence

^MEo —"% Ei e Fo —^—-^ Fi ̂ o (*)

is strictly coexact in Mvs. We know (p is a strict monomorphism of Sns. Let us provethat its range is closed. Assume Xm is a sequence of £'0 such that

(-e(Xm),Uo{Xm)) -^ (y,z)

in EI C FQ. Then, uo(xm) —^ z m FQ and since UQ has closed range, there is x in EQsuch that uo(xm) —^ uo(x). Since UQ is relatively open, Xm -^ x in EQ. Therefore(-e(xm),uo(xm)) —> (-e(x), uo(x)) and since £'1 e FQ is separated, we see that(y^) = (~e(x),uo(x)). Hence (p is a strict monomorphism of ATvs and the sequence(*) is strictly coexact in Sns. It follows that u^ is a strict monomorphism of Sns anit remains to show that it has a closed range. Let Xm be a sequence of E^ such thatui(xm) -> y in Fi. Set ^/ = ^(z,t). This means that

'0(a'm - ̂ -t) —^ 0



in Fi. Since the sequence (*) is strictly exact, there is a sequence Sm of Eo such that

(Xm - Z + e{Sm), -t - Uo{Sm)) -^ 0

in E^ C FQ. It follows that uo(sm) —^ -t in FQ. Hence Sm —^ s in Eo and no^) =-^. Moreover, Xm -^ x = z - e(s) in Ei. Clearly, u^(x) = ^(a-,0) = ^(O,^) +(—e(s),uo(s))) = y and the conclusion follows. D

PROPOSITION 3.2.18. — The canonical inclusion

I : Afvs —^ Snshas a right adjoint

Sep : Sns —> J\fvs.

Moreover, Sep o I = idj^-vs •

Proof. — We define Sep by setting

Sep(E) = E / N

where N == {x € E : pa{x) = 0}. One checks easily that

Hom^(E,J(F)) = RomÊ/N^F)



Sep : Sns —)- A^vs

is strongly right exact and has a left derived functor

LSep:V{Sns) -^ V^JVvs) *c{0,+,-,6).

The functorI : Afvs —^ Sns

is strictly exact and gives rise to a functor

I:V(Sns) -tV\Mvs) * C {0,+,-,&}.

Moreover, I and LSep define quasi-inverse equivalences of categories. In particular,

I : CH^JVvs) -, jCn^Sns)

is an equivalence of categories. (Note that the same result does not hold for KT-iÂfvs)andmi{Sns).)

Proof. — Since any object of Sns is a quotient of an object of the form

®cid

and since objects of this form are clearly separated, one sees easily that Afvs forms aSep-projective subcategory of Sns. The conclusion follows. D



COROLLARY 3.2.20. — The functor

Sep : Sns —> Sns

is left derivable and

LSep:P*(5n5) -^V^Sns) * G {0,+,-,6}

is isomorphic to the identity.

3.2.3. The category W and topological sheaves. — In this section, we fix twouniverses U and V such that U C V.

PROPOSITION 3.2.21. — rfte category

VV = Xndy Snsu

is an elementary quasi-abelian category. It has a canonical closed structure extendingthat of Snsu' For any protective object P ofXndySnsu, the functor

P 0 • : Tndv(Snsu) —^ Tndy{Snsu)

is strongly exact and transforms a projective object into a projective object. In partic-ular,

C1-i(Tndv{Snsu)) wTndy(Cn(Snsu))

is canonically a closed abelian category, the projective objects of which have similarproperties.

Proof. — This is a direct consequence of Proposition 2.1.17, Proposition 2.1.19 andProposition 1.5.4. Q

COROLLARY 3.2.22. — Let P denote the full additive subcategory of Snsy formedby semi-normed spaces of the form

©cid

for some U-set I. Then, we have the canonical equivalence of categories

}VwAdd(P,Abv).

Proof. — This follows from Proposition 2.1.14. D

DEFINITION 3.2.23. — We denote Tc the category formed by locally convex topolog-ical vector spaces and continuous linear maps. For any object E of Tc, we denote byBE the ordered set formed by absolutely convex bounded subsets. For any B e BE,we denote EB the vector subspace of E generated by B endowed with the gaugesemi-norm associated to B.




W : Toy -> W

defined by setting

W{E) = Inn " E B "B^BE

is faithful. Moreover, Rom^(W(E),W(F)) is formed by linear maps from E to Fwhich transform bounded subsets of E into bounded subsets of F. In particular,

Rom^(W(E)^W(F)) =Hom^(E,F)

if E is bornological.

Proof. — This follows directly from the formula

Hom^( lim "EB\ Hm "FB/") ^ ^m Urn Hom^^Fa/).BEBE B ' e B ' F B ^ B E B ' ^ B ' F

D

REMARK 3.2.25. — Let E be an object of Tcy. Through the equivalence of Corol-lary 3.2.22, W{E) corresponds to the functor

P^ Horn ̂ (P,E)

from V to Aby • Note also that

aom^(Q)CÊ)^lÊ)id

where <oo(^; E) denotes the space of bounded families of E which are indexed by I .


W : Tcu -^ W

preserves projective limits. Moreover, an algebraically exact sequence

0 -^ E ' -> E -^ E" -> 0

of FN (resp. DFN) spaces gives rise to the exact sequence

0 -^ W(E') -> W(E) -, W(E") -^ 0

o/W.

Proof. — This follows directly from the preceding remark combined with well-knownresults of functional analysis. D



REMARK 3.2.27. — The preceding result show that a sheaf with values in Tcu giverise to a sheaf with values in W. Note also that the categories of FN and DFNspaces appear as full subcategories of W. Since we may apply to the category Wall the results of the preceding chapter, the cohomological theory of W-sheaves iswell-behaved. Putting all these fact together make us feel that W-sheaves form aconvenient class of topological sheaves for applications to algebraic analysis.


s

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