Grothendieck Tohoku Eng

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  • Some aspects of homological algebra

    Translation of: Sur quelques points dalge`bre homologique

    Alexandre Grothendieck1

    Translated by Marcia L. Barr and Michael Barr

    January 3, 2011

    1The essential content of Chapters 1, 2, and 4, and part of Chapter 3 was developed in the springof 1955 during a seminar in homological algebra at the University of Kansas. Received March 1,1957.

  • Contents

    Introduction iii0.1 Content of the article. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii0.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv0.3 Omissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

    Translators preface vi

    1 Generalities on abelian categories 11.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Additive categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Infinite sums and products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Categories of diagrams and permanence properties . . . . . . . . . . . . . . 101.7 Examples of categories defined by diagram schemes . . . . . . . . . . . . . . 111.8 Inductive and projective limits . . . . . . . . . . . . . . . . . . . . . . . . . 131.9 Generators and cogenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.10 Injective and projective objects . . . . . . . . . . . . . . . . . . . . . . . . . 151.11 Quotient categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    2 Homological algebra in abelian categories 202.1 -functors and -functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Universal -functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Spectral sequences and spectral functors . . . . . . . . . . . . . . . . . . . . 262.5 Resolvent functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    3 Cohomology with coefficients in a sheaf 363.1 General remarks on sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Definition of the Hp(X,F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    i

  • ii CONTENTS

    3.3 Criteria for Acyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Applications to questions of lifting of structure groups . . . . . . . . . . . . 433.5 The exact sequence of a closed subspace . . . . . . . . . . . . . . . . . . . . 493.6 On the cohomological dimension of certain spaces . . . . . . . . . . . . . . . 503.7 The Leray spectral sequence of a continuous function . . . . . . . . . . . . . 543.8 Comparison with Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . 573.9 Acyclicity criteria by the method of covers . . . . . . . . . . . . . . . . . . . 623.10 Passage to the limit in sheaf cohomology . . . . . . . . . . . . . . . . . . . . 65

    4 Ext of sheaves of modules 684.1 The functors HomO(A,B) and HomO(A,B) . . . . . . . . . . . . . . . . . 684.2 The functors ExtpO(X;A,B) and Ext

    pO(A,B) . . . . . . . . . . . . . . . . . 71

    4.3 Case of a constant sheaf of rings . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Case of sheaves with an operator group . . . . . . . . . . . . . . . . . . . . 78

    5 Cohomological study of operator spaces 815.1 Generalities on G-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 The functors Hn(X;G,A) and Hn(G,A) . . . . . . . . . . . . . . . . . . . . 855.3 Case of a discontinuous group of homeomorphisms . . . . . . . . . . . . . . 905.4 Transformation of the first spectral sequence . . . . . . . . . . . . . . . . . 925.5 Computation of the Hn(X;G,A) using covers . . . . . . . . . . . . . . . . . 945.6 The groups ExtnO,G(X;A,B) . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.7 Introduction of families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

  • Introduction

    0.1 Content of the article.

    This work originates from an attempt to take advantage of the formal analogy between thecohomology theory of a space with coefficients in a sheaf [4, 5] and the theory of derivedfunctors of a functor on a category of modules [6], in order to find a common framework toencompass these theories and others.

    This framework is sketched in Chapter 1, whose theme is the same as that of [3]. Thesetwo expositions do not overlap, however, except in 1.4. I have particularly wished to pro-vide usable criteria, with the aid of the concepts of infinite sums and products in abeliancategories, for the existence of sufficiently many injective or projective objects in abeliancategories, without which the essential homological techniques cannot be applied. In ad-dition, for the readers convenience, we will give a thorough exposition of the functoriallanguage (1.1, 1.2, and 1.3). The introduction of additive categories in 1.3 as a preliminaryto abelian categories provides a convenient language (for example to deal with spectralfunctors in Chapter 2).

    Chapter 2 sketches the essential aspects of homological formalism in abelian categories.The publication of [6] has allowed me to be very concise, given that the Cartan-Eilenbergtechniques can be translated without change into the new context. Sections 2.1 and 2.2however, were written so as not to exclude abelian categories that do not contain sufficientlymany injectives or projectives. Later sections are based on resolutions, employing the usualtechniques. Sections 2.4 and 2.5 contain a variety of additional material and are essential forunderstanding what follows them. In particular, Theorem 2.4.1 gives a mechanical methodfor obtaining most known spectral sequences (or, in any case, all those encountered in thiswork).

    In Chapter 3, we redevelop the cohomology theory of a space with coefficients in asheaf, including Lerays classical spectral sequences. The treatment provides additonalflexibility compared with [4, 15], in particular, given that all the essential results are foundwithout any restrictive hypotheses on the relevant spaces, either in this chapter or anylater one, so that the theory also applies to the non-separated spaces that occur in abstractalgebraic geometry or in arithmetical geometry [15, 8]. Conversations with Roger Godement

    iii

  • iv INTRODUCTION

    and Henri Cartan were very valuable for perfecting the theory. In particular, Godementsintroduction of flabby sheaves and soft sheaves, which can useful be substituted for finesheaves in many situations, has turned out to be extremely convenient. A more completedescription, to which we will turn for a variety of details, will be given in a book byGodement in preparation [9].

    Chapter 4 deals with the non-classical question of Ext of sheaves of modules; in particu-lar, it contains a useful spectral sequence that relates global and local Ext. Things get morecomplicated in Chapter 5, in which, in addition, a group G operates on the space X, thesheafO of rings over X and the sheaf ofO-modules under consideration. Specifically, in 5.2,we find what seems to me to be the definitive form of the Cech cohomology theory of spacesacted on, possibly with fixed points, by an abstract group. It is stated by introducing newfunctors Hn(X;G,A) (already implicit in earlier specific cases); we then find two spectralfunctors with remarkable initial terms that converge to it.

    0.2 Applications

    In this article, for want of space, I have been able to provide only very few applicationsof the techniques used (mainly in 3.4 and 3.6), restricting myself to noting only a few inpassing. We indicate the following applications.

    (a) The notion of Ext of sheaves of modules allows the most general formulation known ofSerres algebraic duality theorem: If A is a coherent algebraic sheaf [15] on a projectivealgebraic variety of dimension n without singularities, then the dual of Hp(X,A) iscanonically identified with ExtnpO (X;A

    n), where O (respectively n) is the sheafof germs of regular functions (respectively, of regular n-forms) over X.

    (b) All the formalism developed in Chapters 3, 4, and 5 can apply to abstract algebraicgeometry. I will show elsewhere how it makes possible the extension of various resultsproved by Serre [15, 16, 17] for projective varieties, to complete algebraic varieties.

    (c) It seems that the Hn(X;G,A) are the natural intermediaries for a general theoryof reduced Steenrod powers in sheaves, and the cohomology of symmetric powers ofarbitrary spaces, a theory which also applies in algebraic geometry in characteristicp.

    0.3 Omissions

    To avoid making this memoir overly long, I have said nothing about questions on multi-plicative structures, although they are essential for applying the concepts in Chapters 3, 4,

  • 0.3. OMISSIONS v

    and 5. Note, moreover, that there does not yet seem to be any satisfactory theory of multi-plicative structures in homological algebra that have the necessary generality and simplicity([6, Chapter II] being a striking illustration of this state of affairs).1

    For multiplication

    in sheaf cohomology a satisfactory description can be found in [9]. The reader will noticenumerous other omissions.

    I am happy to express my thanks to Roger Godement, Henri Cartan, and Jean-PierreSerre, whose interest was the indispensable stimulus for the writing of this memoir.

    1Pierre Cartier has recently found a satisfactory general formulation for multiplicative structures inhomological algebra which he will announce himself.

  • Translators preface