Sheaf theory

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A special mathematical tool which provides a unified approach to establishing connections between local and global properties of topological spaces (in particular geometric objects) and which is a powerful method for studying many problems in contemporary algebra, geometry, topology, and analysis.

A pre-sheaf $F$ on a topological space $X$ assigns to each open subset $U\subseteq X$ an Abelian group $F(U)$ (a ring, a module over a ring, etc.) and to every pair of open sets $V\subseteq U$ a homomorphism $F_V^U:F(U)\to F(V)$, such that $F_U^U$ is the identity isomorphism and $F_W^U=F_W^V F_V^U$ for every triple $W\subseteq V\subseteq U$. In other words, a pre-sheaf is a contravariant functor from the category of open subsets of $X$ and their inclusions into the category of groups (rings, etc.) and their homomorphisms. The mappings $F_V^U$ are called restriction homomorphisms (for example, if the elements of the stalk $F(U)$ are functions of some type or other defined on $U$, $F_V^U$ is the restriction of these to the smaller subset). A topology on the set $\def\cF{ {\mathcal F}}\cF = \bigcup_{x\in X}\cF_x$, where $\cF_x$ is, by definition, the direct limit $\lim_{x\in U} F(U)$, defined in the following way: For each $U\subseteq X$ and any $\def\s{\sigma}\s\in F(U)$, the set $S$ consisting of those points of $\cF_x$, $x\in U$, which are images of $\s$ in the definition of $\cF_x$ is declared to be open in $\cF$. In this topology the stalks $\cF_x$ are discrete, the stalk-wise algebraic operations defined on $\cF$ by taking direct limits are continuous and the natural projection $p:\cF\to X$, where $\cF = p^{-1}(x)$, is a local homeomorphism. The space $\cF$ together with the stalk-wise algebraic operations and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over $X$ associated with the pre-sheaf $F$.

Every continuous mapping $s:U\to \cF$ for which $x=ps(x)$ is called a section of $\cF$ over $U$. The section of $\cF$ over $X$ defined by the zeros in $\cF_x$ is called the zero section. If a section $s$ is zero at a point $x$, then $s$ coincides with the zero section in some neighbourhood of $x$; therefore the set of points at which $s$ is not zero (the support of $s$) is closed in $U$.

Let $\def\G{\Gamma}\G(U,\cF)$ (respectively, $\G_\Phi(X,\cF)$, where $\Phi$ is a certain family of closed sets in $X$; in particular, $\G_c(X,\cF)$) be the group (ring, module, etc.) of all sections of $\cF$ over $U$ (respectively, all sections over $X$ with supports in $\Phi$; in particular, sections with compact support). The assignment $U\mapsto\G(U,\cF)$ is a pre-sheaf over $X$, called the pre-sheaf of sections of the sheaf $\cF$. The assignment $\s\mapsto s$ used in defining the topology on $\cF$ also defines homomorphisms $F(U) \to \G(U,\cF)$ which commute with the restrictions to $V\subseteq U$, that is, it defines a homomorphism of pre-sheaves. This homomorphism is an isomorphism provided that the original pre-sheaf $F$ satisfies the requirements:

a) if $\def\l{\lambda}U=\bigcup_\l U_\l$ and $\s, \s'\in F(U)$, then $\s=\s'$ if the restrictions of $\s$ and $\s'$ to each $U_\l$ are equal; and

b) if $U=\bigcup_\l U_\l$ and $\s_\l\in F(U_\l)$ is a collection of elements such that the restrictions of $\s_\l$ and $s_\mu$ to $U_\l\cap U_\mu$ coincide, then there exists a $\s\in F(U)$ which has restriction to each $U_\l$ coinciding with $\s_\l$. The concept of a pre-sheaf satisfying these requirements is equivalent to the concept of the sheaf associated with it, therefore such pre-sheaves are quite often called sheaves also.

A sheaf of the form $X\times G$ (with the evident projection to $X$), where $G$ is a fixed group (ring, etc.), is called a constant sheaf, and is denoted by $G$. A sheaf which is constant in sufficiently small neighbourhoods of every $x\in X$ is called locally constant. The topology of such sheaves is separated (i.e. Hausdorff) if $X$ is a separated space. In more typical situations the topology of $\cF$ can be non-separated even if $X$ is separated (such is the case, for example, for the sheaf of germs of continuous (or differentiable) functions which is generated by the pre-sheaf $F$, where $F(U)$ is the set of continuous (differentiable) functions on $U$; however, the sheaf of germs of analytic functions on a manifold is separated).

Every homomorphism of pre-sheaves $F\to F'$ induces a mapping of the associated sheaves $\cF\to\cF'$, which is a local homeomorphism and maps stalks homomorphically to stalks; such a mapping of sheaves is called a sheaf homomorphism. Mono- and epimorphisms are defined in the standard way. For any sheaf homomorphism $f:\cF'\to\cF$ the image $f(\cF')$ is an open subset of $\cF$, closed with respect to the stalk-wise algebraic operations. Every subset of $\cF$ which satisfies these requirements is called a subsheaf of $\cF$. The quotient sheaf of the sheaf $\cF$ by a subsheaf $\cF'$ is defined as the sheaf $\cF''$ associated with the pre-sheaf $U\mapsto \G(U,\cF)/\G(U,\cF')$; moreover, there is an epimorphism $\cF\to\cF''$, and $\cF_X''\simeq \cF_X/\cF_X'$. For every open $U\subseteq X$ there is a subsheaf in $\cF$, denoted by $\cF_U$, which is the union of $p^{-1}(U)$ with the zero section of $\cF$ over $X$; $\cF_{X\setminus U}$ denotes the corresponding quotient sheaf (whose restriction to $X\setminus U$ coincides with the restriction thereto of $\cF$).

Since it is possible to interpret such ordinary terms as homomorphism, kernel, image, subsheaf, quotient sheaf, etc. for sheaves over $X$ in such a way that these concepts have essentially the same meaning as in algebra, one can consider them from a categorical point of view and apply to sheaf theory the constructions of homological algebra. The resulting category of sheaves over $X$ has the same classical properties as the category of Abelian groups or the category of modules; in particular, one can define for sheaves direct sums, infinite direct products, inductive limits, and other concepts.

The apparatus of sheaf theory has penetrated into various fields of mathematics thanks to the fact that there is a natural definition of the cohomology $\def\H{ {\rm H}}\H^*(X,\cF)$ of a space $X$ with coefficients in a sheaf $\cF$, and this without any kind of restrictions on $X$ (this is essential, for example, in algebraic geometry, where the spaces arising are, as a rule, non-separated) and to the fact that other cohomologies (under certain specific conditions) reduce to a sheaf cohomology, at least in those situations where their application is justified.

To define $\H^*(X,\cF)$ one first constructs the canonical resolution

$$\def\C{ {\rm C}}\C^*(\cF) : 0\to \cF\to\C^0(\cF)\to \C^1(\cF)\to \cdots,$$ where $\C^0(\cF)$ is the sheaf defined by the pre-sheaf $F$ for which $F(U)$ is the group of all (possibly discontinuous) sections of $\cF$ over $U$, so that $\G(U,\C^0(\cF))=F(U),\; \C^1(\cF) = \C^0(\C^0(\cF)/\cF)),\dots,\C^{p+1}(\cF)=\C^0(\C^p(\cF)/{\rm Im}\;\C^{p-1}(\cF)),\dots$. By definition, $\H^p(X,\cF) = \H^p(\G(X,\C^*(\cF)))$ ($\H_\Phi^p(X,\cF))$ is obtained by replacing the symbol $\G$ by $\G_\Phi$). The sheaf $\cF$ itself can be obtained from $C^*(\cF)$ so that $H_\Phi^0(X,\cF)=\G_\Phi(X,\cF)$ (in classical cohomology $\H^0(X,G)$ is the group of locally constant functions on $X$ with values in $G$). The resolution $C^*(\cF) $ is an exact covariant functor of $\cF$: there is a short exact sequence of resolutions corresponding to a short exact "coefficient" sequence $0\to\cF'\to\cF\to\cF''\to 0$. The functor $\G_\Phi$ turns out to be exact on the terms $C^p$, $p\ge0$, of the resolution, therefore there is an exact cohomology sequence

$$\cdots \to \H_\Phi^{p-1}(X,\cF'')\to \H_\Phi^{p}(X,\cF') \to\H_\Phi^p(X,\cF)\to\H_\Phi^{p}(X,\cF'')\to \cdots,$$

corresponding to the indicated coefficient sequence, beginning with $0\to\G_\Phi(X,\cF')\to\G_\Phi(X,\cF)\to\cdots$. The cohomology sequence of a pair $(X,A)$ corresponds to the short exact sequence, $0\to\cF_{X\setminus A}\to\cF\to\cF_A\to 0$ ($A$ is a closed set).

The cohomology groups $\H_\Phi^*(X,\cF)$ have the following "universality" property, which casts light on their meaning: For any other resolution $\def\cL{ {\mathcal L}}\cL^*$ (that is, an exact sequence of sheaves $\cL^q$ beginning with $\cF$) there is a natural "comparison" homomorphism $\H^p(\G_\Phi(X,\cL^*))\to H_\Phi^p(X,\cF)$, which is described in terms of $H_\Phi^p(X,\cL^q)$ by using spectral sequences. An important case is when the sheaves of the resolution are $\Phi$-acyclic, that is, when $H_\Phi^p(X,\cL^q)=0$ for $p\ge 1$; in this case the above homomorphism is an isomorphism. The basic examples of acyclic sheaves are flabby sheaves (for all $U\subseteq X$ the mappings $\G(X,\cL)\to \G(U,\cL)$ are epimorphic) and soft sheaves (any section over a closed set extends to a section over the whole of $X$). The canonical resolution consists of flabby sheaves. If $X$ is a paracompact space, then every flabby sheaf is also soft.

The universality property enables one to compare cohomologies arising in concrete situations with sheaf cohomology (and consequently also with each other), to discern for them the natural bounds within which their application is effective, and also to apply sheaf-theoretic methods to the solution of concrete problems. For example, Aleksandrov–Čech cohomology (cf. Aleksandrov–Čech homology and cohomology) can be defined using cochains obtained from the cochains of a specially selected system of open coverings by taking the direct limit. These cochains turn out to be sections of the sheaves of germs of cochains (defined analogously to the sheaves of germs of functions) constituting a resolution of the group (or even the sheaf) of coefficients, and this sheaf turns out to be soft if the space is paracompact. Thus, for paracompact spaces Aleksandrov–Čech cohomology coincides with a sheaf cohomology. An analogous conclusion holds for Zariski spaces (in particular, for algebraic varieties). Alexander–Spanier cochains also turn out to be sections of the sheaves of a resolution and, moreover, the resolution consists of soft sheaves if $X$ is paracompact, so in this case, in particular, Alexander–Spanier and Aleksandrov–Čech cohomology are naturally isomorphic. In the case of singular cohomology, identification of cochains which coincide on the "small" singular simplices, i.e. subordinated to (arbitrary) open coverings, leads to the so-called localized cochains (giving the same cohomology), which are sections of the sheaves determined by the pre-sheaves of the usual singular cochains. These sheaves turn out to be soft if $X$ is paracompact (if $X$ is hereditarily paracompact, then they are also flabby), but they form a resolution only under the additional requirement that $X$ is weakly locally contractible (in every neighbourhood $U$ of each point $x\in X$ there is a smaller neighbourhood which is contractible to a point inside $U$). A classic example is de Rham's theorem: The cohomology of the complex of differential forms of a differentiable manifold coincides with the usual cohomology with coefficients in the field $\R$ of real numbers (the sheaves of germs of differential forms are soft and form a resolution of $\R$: Sufficiently near to each point each closed differential form is exact).

There are also resolutions corresponding to any open or locally finite closed covering and these enable one to compare the cohomology of $X$ with the cohomology of the coverings (the spectral sequences for coverings). In particular, this gives an isomorphism if $\H^q =0$ for $q\ge 1$ for all elements of the covering and their finite intersections (Leray's theorem). Taking the direct limit with respect to open coverings gives an isomorphism between Aleksandrov–Čech cohomology $\def\cH{ {\mathcal H}}\check{\cH}^*$ and sheaf cohomology, even for non-paracompact $X$, provided that there are sufficiently many small open sets $U$ in $X$ for which $\check{\cH}^q(U,\cF)=0$ when $q\ge 1$ (Cartan's theorem). This means that the cohomologies $\check{\cH}^*$, used in algebraic geometry, with coefficients in coherent sheaves, are also isomorphic to the standard sheaf cohomology $\H^*$.

General constructions ensuring the existence of a comparison homomorphism enable one to compare also the cohomology $\def\cH{ {\mathcal H}}\H^p(X,\cH^q)$ with the hypercohomology $\H^*(\G(X,\cL^*))$ (analogously, $\H_\Phi^p(X,\cH^q)$ with $\H^*(\G_\Phi(X,\cL^*))$) in case $\cL^*$ is any differential sheaf (that is, a sheaf in which for any $q$ the composition $\cL^q\to\cL^{q+2}$ is zero) with $\cL^q$ acyclic, where the $\cH^q$ are the derived sheaves of $\cL^*$ (these are the quotient sheaves of the kernel by the image in each dimension $q$). The corresponding spectral sequences have many applications. Moreover, if $\cH^q=0$ when $q\ge 1$, then $\H^*(\G(X,\cL^*))=\H^*(X,\cH^0)$. For example, if in place of $\cL^*$ a sheaf of chains $\def\cC{ {\mathcal C}}\cC_*$ is taken (the boundary operator lowers the dimension by one, the elements of $\G(U,\cC_*)$ are the chains of the pair $(X,X\setminus U)$, and the stalk $\cH_q^x = \lim_{x\in U}\H_q(X,X\setminus U)=H_q(X,X\setminus x)$), then one obtains the way the homology $\H_*^\Phi(X,G)$ depends on all the possible $\H_\Phi^p(X,\cH_q)$. For a manifold, $\cH_q=0$ when $q>n=\dim X$, and $\H_p^\Phi(X,G)=\H_\Phi^{n-p}(X,\cH_n)$, that is, Poincaré duality holds. If $A$ is an open or closed subset of a locally compact space $X$, then the homology of $A$ is determined by the sections of $\cC_*$ with supports in $A$, and the homology of the pair $(X,A)$ is determined by the sections of the restriction of $\cC_*$ to $X\setminus A$. Conversely (and this is also one of the manifestations of Poincaré duality), if $X\setminus A$ is any flabby resolution for the cohomology, then the restriction of $\cC^*$ to $X\setminus A$ determines the cohomology of $X\setminus A$ and the sections of $\cC^*$ with supports in $A$ determine the cohomology of the pair $(X,X\setminus A)$. Since the sheaves $\cC_*$ are flabby for manifolds, the homology sequence of the pair $(X,A)$ coincides up to an inversion of the numbering with the cohomology sequence of the pair $(X,X\setminus A)$. This means that dualities for manifolds, such as Lefschetz duality $\H_p(X,U,G) = \H^{n-p}(X\setminus U,\cH_n)$, are particular cases of Poincaré duality. It turns out that the duality relations which are not covered by this scheme are corollaries of Poincaré duality and the acyclicity of the manifold in certain dimensions.

Just such a situation arises in the case of a continuous mapping $f:X\to Y$. A resolution for the cohomology of $X$ determines a certain differential sheaf $\cH^*$ on $Y$ for which the stalks $\cH_y^q$ are the direct limits of the cohomology groups $\H^q(f^{-1}(U),\cF)$ with respect to neighbourhoods $U$ of the points $y$ (and for closed mappings $\H_y^q = \H^q(f^{-1}(y),\cF)$), where $\H^*(X,\cF) = \H^*(\G(Y,\cH^*))$. The way $\H^*(X,\cF)$ depends on $\H^p(Y,\cH^q)$ is described by the Leray spectral sequence of the mapping $f$ (a particular case of this is the spectral sequence of a Serre fibration). Acyclic mappings correspond to the case when $\cH^q$ vanishes, thus ensuring that the cohomologies of $X$ and $Y$ with corresponding coefficients are isomorphic (Vietoris' theorem and its generalizations). The general constructions referred to above also give the spectral sequence of a mapping and take into account (along with their cohomological structure) the degree of disconnectedness of pre-images of points; this is especially effective for zero-dimensional or finite-to-one mappings (in the case of coverings it becomes the Cartan spectral sequence). There are also special spectral sequences in categories of $G$-spaces (spaces on which a group $G$ acts).

In sheaf cohomology there is a natural way of defining a multiplicative structure. The existence of special flabby resolutions, of which the mappings are determined by a certain semi-simplicial structure, enables one to give explicit formulas for the products of cochains, analogous to the usual ones. At the same time this also makes it possible to define other cohomology operations in sheaf theory.

The apparatus of sheaf theory has many applications wherever abstract homological methods are essential: in topology (homological and cohomological dimension, local homology and duality, the structure of various classes of continuous mappings, including imbeddings onto dense subsets, and, in particular, compactifications, etc.), in the theory of analytic manifolds (homology and cohomology with coefficients in coherent analytic sheaves and their applications, cohomology and analytic differential forms, homology and analytic flows (the analogue of de Rham's theorem), etc.), and also in abstract algebraic geometry (the cohomology of affine, projective and complete algebraic varieties with coefficients in coherent algebraic sheaves, algebraic Serre duality, algebraic (combinatorial) dimension, etc.).

Some basic notions of sheaf theory and spectral sequences appeared in the work of J. Leray (1945 and later) in connection with the study of homological properties of continuous mappings of locally compact spaces, and he also gave the definition of cohomology (with compact support) with coefficients in a sheaf. A fairly complete account of sheaf theory using resolutions was later given by H. Cartan. The proof of the de Rham theorem given by A. Weil (1947) and the work of J.-P. Serre (in the early 1950's) on algebraic varieties greatly influenced the development of sheaf theory. Cohomology with coefficients in a sheaf was first defined by the Aleksandrov–Čech method. A mature view of sheaf theory could be found by the end of the 1950's in the work of A. Grothendieck [Gr] and R. Godement [Go], where great generality was achieved and the methods were considerably simplified. E.g., it was shown that the category of sheaves over $X$ has a generator (that is, a sheaf $J$ admitting non-zero homomorphisms into any non-zero sheaf; for sheaves of Abelian groups, $J=\sum_{U\subseteq X}\Z_U$), and hence that each sheaf can be imbedded in an injective sheaf (Grothendieck's theorem). This is the reason for the formal analogy between cohomology theory with coefficients in sheaves and the theory of derived functors in the module categories: In the category of sheaves over $X$ there are "enough" injective objects (although, as a rule, there are few projective objects), and therefore one can freely apply all the corresponding techniques of homological algebra; in particular, one can define the cohomology $\H_\Phi^*(X,\cF)$ (without any restriction on $X$) as derived functors of the left exact functor $\G_\Phi(X,\cF)$ (or even as $\def\Ext{ {\rm Ext}}\Ext^*(\Z_X,\cF)$). This also sheds light, for example, on the general nature of such concepts as the cohomological dimension (over $\Z$) of a space, the algebraic dimension of a variety and the global dimension of a ring. The description given by Grothendieck of the spectral sequence for the functor $\Ext$ is essential in algebraic geometry. A much simpler method of constructing injective sheaves was found by Godement. He also showed that to construct a cohomology theory it is entirely sufficient to use his canonical flabby resolution, which, from the point of view of homological algebra, turns out to be simply one of the acyclic resolutions of a sheaf. Godement was the first to apply flabby and soft sheaves (soft sheaves are acyclic only for $X$ paracompact, which explains their use primarily in topology).

Comments

Let $f:X\to Y$ be a continuous mapping between topological spaces and $\cF$ a sheaf on $X$. Then the direct image of $\cF$ under $f$ is the sheaf $f_*\cF$ defined by $(f_*\cF)(U) = \cF(f^{-1}(U))$ for $U$ open in $Y$. The functor $\cF\mapsto f_*(\cF)$ is left-exact on sheaves of modules or Abelian groups. The higher direct images $f_*^q\cF$ of $\cF$ are the right derived functors of $f_*$. The sheaf $f_*^q\cF$ is the sheaf associated to the pre-sheaf $U\mapsto \H^q(f^{-1}(U),\cF)$.

Given a sheaf $\def\cG{ {\mathcal G}}\cG$ over $Y$, let $\pi:\cG\to Y$ be the associated "sheaf space" (so that $\cG(U)=$ sections of $\pi$ over $U\subseteq Y$). Now consider the fibre product (pull-back) $f^*\cG=X\times_Y\cG$. The inverse image sheaf $f^*\cG$ is the sheaf of sections of $f^*\cG\to X$.

The functor $f^*$ is exact (on sheafs of modules or Abelian groups). The two functors are adjoint to one another: ${\rm Hom}(f^*\cG,\cF) \simeq {\rm Hom}(\cG,f_*\cF)$.

A short sequence of sheaves of Abelian groups or modules $0\to\cF'\to\cF\to\cF''\to 0$ over a space $X$ is exact if and only if the corresponding sequence of stalks $0\to\cF_x'\to\cF_x\to\cF_x''\to 0$ is exact for every $x\in X$.

Grothendieck and his school vastly generalized sheaf theory, from sheaves on a space to the more general notion of sheaves on a site and that of a topos (cf. [Ar]–[Gr3]).

For the theory of coherent sheaves on algebraic varieties see also the fundamental paper [Gr2]. For the theory of sheaves in the étale topology and for $l$-adic sheaves see [Gr3], [De]–[FrKi].

For a detailed history of the origins of sheaf theory, see [Gr2]. The article above focuses almost entirely on the use of sheaf theory in cohomology theory, but it does have many other applications: for example, in the representation theory of rings and other algebraic systems (see, for example, [Mu], [BoBo]), and in logic, in the provision of models for constructive mathematics (see [FoSc], and also [MaRe], [Ti]).

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