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Sheaf (mathematics)

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From Wikipedia, the free encyclopedia

Tool to track locally defined data attached to the open sets of a topological space

This article is about sheaves ontopological spaces. For sheaves on a site, seeGrothendieck topology andTopos.

Look upsheaf in Wiktionary, the free dictionary.

Inmathematics, asheaf (pl.:sheaves) is a tool for systematically tracking data (such as sets,abelian groups,rings) attached to theopen sets of atopological space and defined locally with regard to them. For example, for each open set, the data could be the ring ofcontinuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open setscovering the original open set (intuitively, every datum is the sum of its constituent data).

The field of mathematics that studies sheaves is calledsheaf theory.

Sheaves are understood conceptually as general and abstractobjects. Their precise definition is rather technical. They are specifically defined assheaves of sets or assheaves of rings, for example, depending on the type of data assigned to the open sets.

There are alsomaps (ormorphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves ofabelian groups) with their morphisms on a fixed topological space form acategory. On the other hand, to eachcontinuous map there is associated both adirect image functor, taking sheaves and their morphisms on thedomain to sheaves and morphisms on thecodomain, and aninverse image functor operating in the opposite direction. Thesefunctors, and certain variants of them, are essential parts of sheaf theory.

Due to their general nature and versatility, sheaves have several applications in topology and especially inalgebraic anddifferential geometry. First, geometric structures such as that of adifferentiable manifold or ascheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such asvector bundles ordivisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very generalcohomology theory, which encompasses also the "usual" topological cohomology theories such assingular cohomology. Especially in algebraic geometry and the theory ofcomplex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory ofD-modules, which provide applications to the theory ofdifferential equations. In addition, generalisations of sheaves to more general settings than topological spaces, such as the notion of a sheaf on a category with respect to someGrothendieck topology, have provided applications tomathematical logic and tonumber theory.

Definitions and examples

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In many mathematical branches, several structures defined on atopological space $X {\displaystyle X}$ (e.g., adifferentiable manifold) can be naturallylocalised orrestricted toopen subsets $U\subseteq X$ : typical examples includecontinuous real-valued orcomplex-valued functions, $n {\displaystyle n}$ -timesdifferentiable (real-valued or complex-valued) functions,bounded real-valued functions,vector fields, andsections of anyvector bundle on the space. The ability to restrict data to smaller open subsets gives rise to the concept of presheaves. Roughly speaking, sheaves are then those presheaves, where local data can be glued to global data.

Presheaves

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Sheaves

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Given a presheaf, a natural question to ask is to what extent its sections over an open set $U {\displaystyle U}$ are specified by their restrictions to open subsets of $U {\displaystyle U}$ . Asheaf is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.

Axiomatically, asheaf is a presheaf that satisfies both of the following axioms:

(Locality) Suppose $U {\displaystyle U}$ is an open set, $\{U_{i}\}_{i\in I}$ is an open cover of $U {\displaystyle U}$ with $U_{i}\subseteq U$ for all $i\in I$ , and $s,t\in {\mathcal {F}}(U)$ are sections. If $s|_{U_{i}}=t|_{U_{i}}$ for all $i\in I$ , then $s=t$ .
(Gluing) Suppose $U {\displaystyle U}$ is an open set, $\{U_{i}\}_{i\in I}$ is an open cover of $U {\displaystyle U}$ with $U_{i}\subseteq U$ for all $i\in I$ , and $\{s_{i}\in {\mathcal {F}}(U_{i})\}_{i\in I}$ is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if $s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}$ for all $i,j\in I$ , then there exists a section $s\in {\mathcal {F}}(U)$ such that $s|_{U_{i}}=s_{i}$ for all $i\in I$ .^[1]

Sections over two opens of the two-point space.
Gluing compatible local sections to a section over the union.

In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that ${\textstyle \bigcup _{i\in I}U_{i}=U}$ .

The section $s {\displaystyle s}$ whose existence is guaranteed by axiom 2 is called thegluing,concatenation, orcollation of the sections $s_{i}$ . By axiom 1 it is unique. Sections $s_{i}$ and $s_{j}$ satisfying the agreement precondition of axiom 2 are often calledcompatible ; thus axioms 1 and 2 together state thatany collection of pairwise compatible sections can be uniquely glued together. Aseparated presheaf, ormonopresheaf, is a presheaf satisfying axiom 1.^[2]

The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions $f_{i}:U_{i}\to \mathbb {R}$ which agree on the intersections $U_{i}\cap U_{j}$ , there is a unique continuous function $f:U\to \mathbb {R}$ whose restriction equals the $f_{i}$ . By contrast, the constant presheaf is usuallynot a sheaf as it fails to satisfy the locality axiom on theempty set (this is explained in more detail atconstant sheaf).

Presheaves and sheaves are typically denoted by capital letters, $F {\displaystyle F}$ being particularly common, presumably for theFrench word for sheaf,faisceau. Use of calligraphic letters such as ${\mathcal {F}}$ is also common.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of abasis for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. This observation is used to construct another example which is crucial in algebraic geometry, namelyquasi-coherent sheaves. Here the topological space in question is thespectrum of a commutative ring $R {\displaystyle R}$ , whose points are theprime ideals ${\mathfrak {p}}$ in $R {\displaystyle R}$ . The open sets $D_{f}:=\{{\mathfrak {p}}\subseteq R,f\notin {\mathfrak {p}}\}$ form a basis for theZariski topology on this space. Given an $R {\displaystyle R}$ -module $M {\displaystyle M}$ , there is a sheaf, denoted by ${\tilde {M}}$ on the $\operatorname {Spec} R$ , that satisfies

{\tilde {M}}(D_{f}):=M[1/f],

thelocalization of

M {\displaystyle M}

f {\displaystyle f}

There is another characterization of sheaves that is equivalent to the previously discussed.A presheaf ${\mathcal {F}}$ is a sheafif and only if for any open $U {\displaystyle U}$ and any open cover $\{U_{a}\}$ of $U {\displaystyle U}$ , ${\mathcal {F}}(U)$ is the fibre product ${\mathcal {F}}(U)\cong {\mathcal {F}}(U_{a})\times _{{\mathcal {F}}(U_{a}\cap U_{b})}{\mathcal {F}}(U_{b})$ . This characterization is useful in construction of sheaves, for example, if ${\mathcal {F}},{\mathcal {G}}$ areabelian sheaves, then the kernel of sheaves morphism ${\mathcal {F}}\to {\mathcal {G}}$ is a sheaf, since projective limits commutes with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limits do not necessarily commute with projective limits. One way to fix this is to consider Noetherian topological spaces; all open sets are compact so that the cokernel is a sheaf, since finite projective limits commutes with inductive limits.

Further examples

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Sheaf of sections of a continuous map

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Any continuous map $f:Y\to X$ of topological spaces determines a sheaf $\Gamma (Y/X)$ on $X {\displaystyle X}$ by setting

\Gamma (Y/X)(U)=\{s:U\to Y,f\circ s=\operatorname {id} _{U}\}.

Any such $s {\displaystyle s}$ is commonly called asection of $f {\displaystyle f}$ , and this example is the reason why the elements in ${\mathcal {F}}(U)$ are generally called sections. This construction is especially important when $f {\displaystyle f}$ is the projection of afiber bundle onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of thetrivial bundle.

Another example: the sheaf of sections of

\mathbb {C} {\stackrel {\exp }{\longrightarrow }}\mathbb {C} \setminus \{0\}

is the sheaf which assigns to any $U\subseteq \mathbb {C} \setminus \{0\}$ the set of branches of thecomplex logarithm on $U {\displaystyle U}$ .

Given a point $x {\displaystyle x}$ and an abelian group $S {\displaystyle S}$ , the skyscraper sheaf $S_{x}$ is defined as follows: if $U {\displaystyle U}$ is an open set containing $x {\displaystyle x}$ , then $S_{x}(U)=S$ . If $U {\displaystyle U}$ does not contain $x {\displaystyle x}$ , then $S_{x}(U)=0$ , thetrivial group. The restriction maps are either the identity on $S {\displaystyle S}$ , if both open sets contain $x {\displaystyle x}$ , or the zero map otherwise.

Sheaves on manifolds

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On an $n {\displaystyle n}$ -dimensional $C^{k}$ -manifold $M {\displaystyle M}$ , there are a number of important sheaves, such as the sheaf of $j {\displaystyle j}$ -times continuously differentiable functions ${\mathcal {O}}_{M}^{j}$ (with $j\leq k$ ). Its sections on some open $U {\displaystyle U}$ are the $C^{j}$ -functions $U\to \mathbb {R}$ . For $j=k$ , this sheaf is called thestructure sheaf and is denoted ${\mathcal {O}}_{M}$ . The nonzero $C^{k}$ functions also form a sheaf, denoted ${\mathcal {O}}_{X}^{\times }$ .Differential forms (of degree $p {\displaystyle p}$ ) also form a sheaf $\Omega _{M}^{p}$ . In all these examples, the restriction morphisms are given by restricting functions or forms.

The assignment sending $U {\displaystyle U}$ to the compactly supported functions on $U {\displaystyle U}$ is not a sheaf, since there is, in general, no way to preserve this property by passing to a smaller open subset. Instead, this forms acosheaf, adual concept where the restriction maps go in the opposite direction than with sheaves.^[3] However, taking thedual of these vector spaces does give a sheaf, the sheaf ofdistributions.

Presheaves that are not sheaves

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In addition to the constant presheaf mentioned above, which is usually not a sheaf, there are further examples of presheaves that are not sheaves:

Let $X {\displaystyle X}$ be thetwo-point topological space $\{x,y\}$ with the discrete topology. Define a presheaf $F {\displaystyle F}$ as follows: $F(\varnothing )=\{\varnothing \},\ F(\{x\})=\mathbb {R} ,\ F(\{y\})=\mathbb {R} ,\ F(\{x,y\})=\mathbb {R} \times \mathbb {R} \times \mathbb {R}$ The restriction map $F(\{x,y\})\to F(\{x\})$ is the projection of $\mathbb {R} \times \mathbb {R} \times \mathbb {R}$ onto its first coordinate, and the restriction map $F(\{x,y\})\to F(\{y\})$ is the projection of $\mathbb {R} \times \mathbb {R} \times \mathbb {R}$ onto its second coordinate. $F {\displaystyle F}$ is a presheaf that is not separated: a global section is determined by three numbers, but the values of that section over $\{x\}$ and $\{y\}$ determine only two of those numbers. So while we can glue any two sections over $\{x\}$ and $\{y\}$ , we cannot glue them uniquely.
Let $X=\mathbb {R}$ be thereal line, and let $F(U)$ be the set ofbounded continuous functions on $U {\displaystyle U}$ . This is not a sheaf because it is not always possible to glue. For example, let $U_{i}$ be the set of all $x {\displaystyle x}$ such that $|x|<i$ . The identity function $f(x)=x$ is bounded on each $U_{i}$ . Consequently, we get a section $s_{i}$ on $U_{i}$ . However, these sections do not glue, because the function $f {\displaystyle f}$ is not bounded on the real line. Consequently $F {\displaystyle F}$ is a presheaf, but not a sheaf. In fact, $F {\displaystyle F}$ is separated because it is a sub-presheaf of the sheaf of continuous functions.

Motivating sheaves from complex analytic spaces and algebraic geometry

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One of the historical motivations for sheaves have come from studyingcomplex manifolds,^[4]complex analytic geometry,^[5] andscheme theory fromalgebraic geometry. This is because in all of the previous cases, we consider a topological space $X {\displaystyle X}$ together with a structure sheaf ${\mathcal {O}}$ giving it the structure of a complex manifold, complex analytic space, or scheme. This perspective of equipping a topological space with a sheaf is essential to the theory of locally ringed spaces (see below).

Technical challenges with complex manifolds

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One of the main historical motivations for introducing sheaves was constructing a device which keeps track ofholomorphic functions oncomplex manifolds. For example, on acompact complex manifold $X {\displaystyle X}$ (likecomplex projective space or thevanishing locus in projective space of ahomogeneous polynomial), theonly holomorphic functions

$f:X\to \mathbb {C}$

are the constant functions.^[6]^[7] This means there exist two compact complex manifolds $X, X^{'} {\displaystyle X,X'}$ which are not isomorphic, but nevertheless their rings of global holomorphic functions, denoted ${\mathcal {H}}(X),{\mathcal {H}}(X')$ , are isomorphic. Contrast this withsmooth manifolds where every manifold $M {\displaystyle M}$ can be embedded inside some $\mathbb {R} ^{n}$ , hence its ring of smooth functions $C^{\infty }(M)$ comes from restricting the smooth functions from $C^{\infty }(\mathbb {R} ^{n})$ , of which there exist plenty.

Another complexity when considering the ring of holomorphic functions on a complex manifold $X {\displaystyle X}$ is given a small enough open set $U\subseteq X$ , the holomorphic functions will be isomorphic to ${\mathcal {H}}(U)\cong {\mathcal {H}}(\mathbb {C} ^{n})$ . Sheaves are a direct tool for dealing with this complexity since they make it possible to keep track of the holomorphic structure on the underlying topological space of $X {\displaystyle X}$ on arbitrary open subsets $U\subseteq X$ . This means as $U {\displaystyle U}$ becomes more complex topologically, the ring ${\mathcal {H}}(U)$ can be expressed from gluing the ${\mathcal {H}}(U_{i})$ . Note that sometimes this sheaf is denoted ${\mathcal {O}}(-)$ or just ${\mathcal {O}}$ , or even ${\mathcal {O}}_{X}$ when we want to emphasize the space the structure sheaf is associated to.

Tracking submanifolds with sheaves

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Another common example of sheaves can be constructed by considering a complex submanifold $Y\hookrightarrow X$ . There is an associated sheaf ${\mathcal {O}}_{Y}$ which takes an open subset $U\subseteq X$ and gives the ring of holomorphic functions on $U\cap Y$ . This kind of formalism was found to be extremely powerful and motivates a lot ofhomological algebra such assheaf cohomology since anintersection theory can be built using these kinds of sheaves from the Serre intersection formula.

Operations with sheaves

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Morphisms

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Morphisms of sheaves are, roughly speaking, analogous to functions between them. In contrast to a function between sets, which is simply an assignment of outputs to inputs, morphisms of sheaves are also required to be compatible with the local–global structures of the underlying sheaves. This idea is made precise in the following definition.

Let ${\mathcal {F}}$ and ${\mathcal {G}}$ be two sheaves of sets (respectively abelian groups, rings, etc.) on $X {\displaystyle X}$ . Amorphism $\varphi :{\mathcal {F}}\to {\mathcal {G}}$ consists of a morphism $\varphi _{U}:{\mathcal {F}}(U)\to {\mathcal {G}}(U)$ of sets (respectively abelian groups, rings, etc.) for each open set $U {\displaystyle U}$ of $X {\displaystyle X}$ , subject to the condition that this morphism is compatible with restrictions. In other words, for every open subset $V {\displaystyle V}$ of an open set $U {\displaystyle U}$ , the following diagram iscommutative.

{\begin{array}{rcl}{\mathcal {F}}(U)&\xrightarrow {\quad \varphi _{U}\quad } &{\mathcal {G}}(U)\\r_{V}^{U}{\Biggl \downarrow }&&{\Biggl \downarrow }{r'}_{V}^{U}\\{\mathcal {F}}(V)&{\xrightarrow[{\quad \varphi _{V}\quad }]{}}&{\mathcal {G}}(V)\end{array}}

For example, taking the derivative gives a morphism of sheaves on $\mathbb {R}$ , ${\frac {\mathrm {d} }{\mathrm {d} x}}\colon {\mathcal {O}}_{\mathbb {R} }^{n}\to {\mathcal {O}}_{\mathbb {R} }^{n-1}.$ Indeed, given an ( $n {\displaystyle n}$ -times continuously differentiable) function $f:U\to \mathbb {R}$ (with $U {\displaystyle U}$ in $\mathbb {R}$ open), the restriction (to a smaller open subset $V {\displaystyle V}$ ) of its derivative equals the derivative of $f|_{V}$ .

With this notion of morphism, sheaves of sets (respectively abelian groups, rings, etc.) on a fixed topological space $X {\displaystyle X}$ form acategory. The general categorical notions ofmono-,epi- andisomorphisms can therefore be applied to sheaves.

In fact, from the point of view of category theory, the category of sheaves over a (small) category $C {\displaystyle C}$ with values in another category $D {\displaystyle D}$ is a full subcategory of the category ofpresheaves over $C {\displaystyle C}$ with values in $D {\displaystyle D}$ , which is simply the category $D^{C^{\text{op}}}$ of contravariant functors from $C {\displaystyle C}$ to $D {\displaystyle D}$ with natural transformations between them as morphisms: the notion of morphism defined above can simply be stated as $\varphi$ being a natural transformation between the two sheaves seen as functors.

A morphism $\varphi \colon {\mathcal {F}}\rightarrow {\mathcal {G}}$ of sheaves on $X {\displaystyle X}$ is an isomorphism (respectively monomorphism) if and only if for every open set $U\subseteq X$ , we have an isomorphism ${\mathcal {F}}(U)\approx {\mathcal {G}}(U)$ which is natural with respect to the restriction maps. These statements give examples of how to work with sheaves using local information, but it's important to note that we cannot check if a morphism of sheaves is an epimorphism in the same manner. Indeed the statement that maps on the level of open sets $\varphi _{U}\colon {\mathcal {F}}(U)\rightarrow {\mathcal {G}}(U)$ are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor—or equivalently, to non-triviality ofsheaf cohomology.

Stalks of a sheaf

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Main article:Stalk (sheaf)

Two vertical stalks above two base points, with germs marked along each stalk. — Stalks and germs for a constant sheaf on a discrete two-point space.

Thestalk ${\mathcal {F}}_{x}$ of a sheaf ${\mathcal {F}}$ captures the properties of a sheaf "around" a point $x\in X$ , generalizing thegerms of functions.Here, "around" means that, conceptually speaking, one looks at smaller and smallerneighborhoods of the point. Of course, no single neighborhood will be small enough, which requires considering a limit of some sort. More precisely, the stalk is defined by

{\mathcal {F}}_{x}=\varinjlim _{U\ni x}{\mathcal {F}}(U),

thedirect limit being over all open subsets of $X {\displaystyle X}$ containing the given point $x {\displaystyle x}$ . In other words, an element of the stalk is given by a section over some open neighborhood of $x {\displaystyle x}$ , and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

The natural morphism ${\mathcal {F}}(U)\to {\mathcal {F}}_{x}$ takes a section $s {\displaystyle s}$ in ${\mathcal {F}}(U)$ to itsgerm $s_{x}$ at $x {\displaystyle x}$ . This generalises the usual definition of agerm.

In many situations, knowing the stalks of a sheaf is enough to control the sheaf itself. For example, whether or not a morphism of sheaves is a monomorphism, epimorphism, or isomorphism can be tested on the stalks. In this sense, a sheaf is determined by its stalks, which are a local data. By contrast, theglobal information present in a sheaf, i.e., theglobal sections, i.e., the sections ${\mathcal {F}}(X)$ on the whole space $X {\displaystyle X}$ , typically carry less information. For example, for acompact complex manifold $X {\displaystyle X}$ , the global sections of the sheaf of holomorphic functions are just $\mathbb {C}$ , since any holomorphic function

X\to \mathbb {C}

is constant byLiouville's theorem.^[6]

Turning a presheaf into a sheaf

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It is frequently useful to take the data contained in a presheaf and to express it as a sheaf. It turns out that there is a best possible way to do this. It takes a presheaf ${\mathcal {F}}$ and produces a new sheaf $a{\mathcal {F}}$ called thesheafification orsheaf associated to the presheaf ${\mathcal {F}}$ . For example, the sheafification of the constant presheaf (see above) is called theconstant sheaf. Despite its name, its sections arelocally constant functions.

The sheaf $a{\mathcal {F}}$ can be constructed using theétalé space $E {\displaystyle E}$ of ${\mathcal {F}}$ , namely as the sheaf of sections of the map

E\to X.

Another construction of the sheaf $a{\mathcal {F}}$ proceeds by means of a functor $L {\displaystyle L}$ from presheaves to presheaves that gradually improves the properties of a presheaf: for any presheaf ${\mathcal {F}}$ , $L{\mathcal {F}}$ is a separated presheaf, and for any separated presheaf ${\mathcal {F}}$ , $L{\mathcal {F}}$ is a sheaf. The associated sheaf $a{\mathcal {F}}$ is given by $LL{\mathcal {F}}$ .^[8]

The idea that the sheaf $a{\mathcal {F}}$ is the best possible approximation to ${\mathcal {F}}$ by a sheaf is made precise using the followinguniversal property: there is a natural morphism of presheaves $i\colon {\mathcal {F}}\to a{\mathcal {F}}$ so that for any sheaf ${\mathcal {G}}$ and any morphism of presheaves $f\colon {\mathcal {F}}\to {\mathcal {G}}$ , there is a unique morphism of sheaves ${\tilde {f}}\colon a{\mathcal {F}}\rightarrow {\mathcal {G}}$ such that $f={\tilde {f}}i$ . In fact, $a {\displaystyle a}$ is the leftadjoint functor to the inclusion functor (orforgetful functor) from the category of sheaves to the category of presheaves, and $i {\displaystyle i}$ is theunit of the adjunction. In this way, the category of sheaves turns into aGiraud subcategory of presheaves. This categorical situation is the reason why the sheafification functor appears in constructing cokernels of sheaf morphisms or tensor products of sheaves, but not for kernels, say.

Subsheaves, quotient sheaves

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If $K {\displaystyle K}$ is asubsheaf of a sheaf $F {\displaystyle F}$ of abelian groups, then thequotient sheaf $Q {\displaystyle Q}$ is the sheaf associated to the presheaf $U\mapsto F(U)/K(U)$ ; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;

0\to K\to F\to Q\to 0.

(this is also called asheaf extension.)

Let $F, G {\displaystyle F,G}$ be sheaves of abelian groups. The set $\operatorname {Hom} (F,G)$ of morphisms of sheaves from $F {\displaystyle F}$ to $G {\displaystyle G}$ forms an abelian group (by the abelian group structure of $G {\displaystyle G}$ ). Thesheaf hom of $F {\displaystyle F}$ and $G {\displaystyle G}$ , denoted by,

{\mathcal {Hom}}(F,G)

is the sheaf of abelian groups $U\mapsto \operatorname {Hom} (F|_{U},G|_{U})$ where $F|_{U}$ is the sheaf on $U {\displaystyle U}$ given by $(F|_{U})(V)=F(V)$ (note sheafification is not needed here). The direct sum of $F {\displaystyle F}$ and $G {\displaystyle G}$ is the sheaf given by $U\mapsto F(U)\oplus G(U)$ , and the tensor product of $F {\displaystyle F}$ and $G {\displaystyle G}$ is the sheaf associated to the presheaf $U\mapsto F(U)\otimes G(U)$ .

All of these operations extend tosheaves of modules over asheaf of rings $A {\displaystyle A}$ ; the above is the special case when $A {\displaystyle A}$ is theconstant sheaf ${\underline {\mathbf {Z} }}$ .

Basic functoriality

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Main article:Image functors for sheaves

Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map $f:X\to Y$ between two topological spaces, pushforward and pullback relate sheaves on $X {\displaystyle X}$ to those on $Y {\displaystyle Y}$ and vice versa.

Direct image

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The pushforward (also known asdirect image) of a sheaf ${\mathcal {F}}$ on $X {\displaystyle X}$ is the sheaf defined by

(f_{*}{\mathcal {F}})(V)={\mathcal {F}}(f^{-1}(V)).

Here $V {\displaystyle V}$ is an open subset of $Y {\displaystyle Y}$ , so that its preimage is open in $X {\displaystyle X}$ by the continuity of $f {\displaystyle f}$ . This construction recovers the skyscraper sheaf $S_{x}$ mentioned above:

S_{x}=i_{*}(S)

where $i:\{x\}\to X$ is the inclusion, and $S {\displaystyle S}$ is regarded as a sheaf on thesingleton by $S(\{*\})=S,S(\emptyset )=\emptyset$ .

For a map betweenlocally compact spaces, thedirect image with compact support is a subsheaf of the direct image.^[9] By definition, $(f_{!}{\mathcal {F}})(V)$ consists of those $s\in {\mathcal {F}}(f^{-1}(V))$ whosesupport is mappedproperly. If $f {\displaystyle f}$ is proper itself, then $f_{!}{\mathcal {F}}=f_{*}{\mathcal {F}}$ , but in general they disagree.

Inverse image

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The pullback orinverse image goes the other way: it produces a sheaf on $X {\displaystyle X}$ , denoted $f^{-1}{\mathcal {G}}$ out of a sheaf ${\mathcal {G}}$ on $Y {\displaystyle Y}$ . If $f {\displaystyle f}$ is the inclusion of an open subset, then the inverse image is just a restriction, i.e., it is given by $(f^{-1}{\mathcal {G}})(U)={\mathcal {G}}(U)$ for an open $U {\displaystyle U}$ in $X {\displaystyle X}$ . A sheaf ${\mathcal {F}}$ (on some space $X {\displaystyle X}$ ) is calledlocally constant if $X=\bigcup _{i\in I}U_{i}$ by some open subsets $U_{i}$ such that the restriction of ${\mathcal {F}}$ to all these open subsets is constant. On a wide range of topological spaces $X {\displaystyle X}$ , such sheaves areequivalent torepresentations of thefundamental group $\pi _{1}(X)$ .

For general maps $f {\displaystyle f}$ , the definition of $f^{-1}{\mathcal {G}}$ is more involved; it is detailed atinverse image functor. The stalk is an essential special case of the pullback in view of a natural identification, where $i {\displaystyle i}$ is as above:

{\mathcal {G}}_{x}=i^{-1}{\mathcal {G}}(\{x\}).

More generally, stalks satisfy $(f^{-1}{\mathcal {G}})_{x}={\mathcal {G}}_{f(x)}$ .

Extension by zero

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"Extension by zero" redirects here. For uses in analysis, seeSobolev space § Extension by zero, andExtension of a function.

For the inclusion $j:U\to X$ of an open subset, theextension by zero $j_{!}{\mathcal {F}}$ (pronounced "j lowershriek of F") of a sheaf ${\mathcal {F}}$ of abelian groups on $U {\displaystyle U}$ is the sheafification of the presheaf defined by

V\mapsto {\mathcal {F}}(V)

V\subseteq U

and

V\mapsto 0

otherwise.

For a sheaf ${\mathcal {G}}$ on $X {\displaystyle X}$ , this construction is in a sense complementary to $i_{*}$ , where $i:X\setminus U\to X$ is the inclusion of the complement of $U {\displaystyle U}$ :

(j_{!}j^{*}{\mathcal {G}})_{x}={\mathcal {G}}_{x}

for

x {\displaystyle x}

U {\displaystyle U}

, and the stalk is zero otherwise, while

(i_{*}i^{*}{\mathcal {G}})_{x}=0

for

x {\displaystyle x}

U {\displaystyle U}

, and equals

{\mathcal {G}}_{x}

otherwise.

More generally, if $A\subset X$ is alocally closed subset, then there exists an open $U {\displaystyle U}$ of $X {\displaystyle X}$ containing $A {\displaystyle A}$ such that $A {\displaystyle A}$ is closed in $U {\displaystyle U}$ . Let $f:A\to U$ and $j:U\to X$ be the natural inclusions. Then theextension by zero of a sheaf ${\mathcal {F}}$ on $A {\displaystyle A}$ is defined by $j_{!}f_{*}F$ .

Due to its nice behavior on stalks, the extension by zero functor is useful for reducing sheaf-theoretic questions on $X {\displaystyle X}$ to ones on the strata of astratification, i.e., a decomposition of $X {\displaystyle X}$ into smaller, locally closed subsets.

Complements

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Sheaves in more general categories

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In addition to (pre-)sheaves as introduced above, where ${\mathcal {F}}(U)$ is merely a set, it is in many cases important to keep track of additional structure on these sections. For example, the sections of the sheaf of continuous functions naturally form a realvector space, and restriction is alinear map between these vector spaces.

Presheaves with values in an arbitrary category $C {\displaystyle C}$ are defined by first considering the category of open sets on $X {\displaystyle X}$ to be theposetal category $O(X)$ whose objects are the open sets of $X {\displaystyle X}$ and whose morphisms are inclusions. Then a $C {\displaystyle C}$ -valued presheaf on $X {\displaystyle X}$ is the same as acontravariant functor from $O(X)$ to $C {\displaystyle C}$ . Morphisms in this category of functors, also known asnatural transformations, are the same as the morphisms defined above, as can be seen by unraveling the definitions.

If the target category $C {\displaystyle C}$ admits alllimits, a $C {\displaystyle C}$ -valued presheaf is a sheaf if the following diagram is anequalizer for every open cover ${\mathcal {U}}=\{U_{i}\}_{i\in I}$ of any open set $U {\displaystyle U}$ :

F(U)\rightarrow \prod _{i}F(U_{i}){{{} \atop \longrightarrow } \atop {\longrightarrow  \atop {}}}\prod _{i,j}F(U_{i}\cap U_{j}).

Here the first map is the product of the restriction maps

\operatorname {res} _{U_{i},U}\colon F(U)\rightarrow F(U_{i})

and the pair of arrows the products of the two sets of restrictions

\operatorname {res} _{U_{i}\cap U_{j},U_{i}}\colon F(U_{i})\rightarrow F(U_{i}\cap U_{j})

and

\operatorname {res} _{U_{i}\cap U_{j},U_{j}}\colon F(U_{j})\rightarrow F(U_{i}\cap U_{j}).

If $C {\displaystyle C}$ is anabelian category, this condition can also be rephrased by requiring that there is anexact sequence

0\to F(U)\to \prod _{i}F(U_{i})\xrightarrow {\operatorname {res} _{U_{i}\cap U_{j},U_{i}}-\operatorname {res} _{U_{i}\cap U_{j},U_{j}}} \prod _{i,j}F(U_{i}\cap U_{j}).

A particular case of this sheaf condition occurs for $U {\displaystyle U}$ being the empty set, and the index set $I {\displaystyle I}$ also being empty. In this case, the sheaf condition requires ${\mathcal {F}}(\emptyset )$ to be theterminal object in $C {\displaystyle C}$ .

Ringed spaces and sheaves of modules

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Main articles:Ringed space andSheaf of modules

In several geometrical disciplines, includingalgebraic geometry anddifferential geometry, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by ${\mathcal {O}}_{X}$ . Such a pair $(X,{\mathcal {O}}_{X})$ is called aringed space. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks ${\mathcal {O}}_{X,x}$ of the structure sheaf arelocal rings, in which case the pair is called alocally ringed space.

For example, an $n {\displaystyle n}$ -dimensional $C^{k}$ manifold $M {\displaystyle M}$ is a locally ringed space whose structure sheaf consists of $C^{k}$ -functions on the open subsets of $M {\displaystyle M}$ . The property of being alocally ringed space translates into the fact that such a function, which is nonzero at a point $x {\displaystyle x}$ , is also non-zero on a sufficiently small open neighborhood of $x {\displaystyle x}$ . Some authors actuallydefine real (or complex) manifolds to be locally ringed spaces that are locally isomorphic to the pair consisting of an open subset of $\mathbb {R} ^{n}$ (respectively $\mathbb {C} ^{n}$ ) together with the sheaf of $C^{k}$ (respectively holomorphic) functions.^[10] Similarly,schemes, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to thespectrum of a ring.

Given a ringed space, asheaf of modules is a sheaf ${\mathcal {M}}$ such that on every open set $U {\displaystyle U}$ of $X {\displaystyle X}$ , ${\mathcal {M}}(U)$ is an ${\mathcal {O}}_{X}(U)$ -module and for every inclusion of open sets $V\subseteq U$ , the restriction map ${\mathcal {M}}(U)\to {\mathcal {M}}(V)$ is compatible with the restriction map ${\mathcal {O}}(U)\to {\mathcal {O}}(V)$ : the restriction of $f s {\displaystyle fs}$ is the restriction of $f {\displaystyle f}$ times that of $s {\displaystyle s}$ for any $f {\displaystyle f}$ in ${\mathcal {O}}(U)$ and $s {\displaystyle s}$ in ${\mathcal {M}}(U)$ .

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence betweenvector bundles andlocally free sheaves of ${\mathcal {O}}_{X}$ -modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where ${\mathcal {O}}$ consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are $D {\displaystyle D}$ -modules, that is, modules over the sheaf ofdifferential operators. On any topological space, modules over the constant sheaf ${\underline {\mathbf {Z} }}$ are the same assheaves of abelian groups in the sense above.

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted $f^{*}$ and it is distinct from $f^{-1}$ . Seeinverse image functor.

Finiteness conditions for sheaves of modules

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Finiteness conditions for module overcommutative rings give rise to similar finiteness conditions for sheaves of modules: ${\mathcal {M}}$ is calledfinitely generated (respectivelyfinitely presented) if, for every point $x {\displaystyle x}$ of $X {\displaystyle X}$ , there exists an open neighborhood $U {\displaystyle U}$ of $x {\displaystyle x}$ , a natural number $n {\displaystyle n}$ (possibly depending on $U {\displaystyle U}$ ), and a surjective morphism of sheaves ${\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}$ (respectively, in addition a natural number $m {\displaystyle m}$ , and an exact sequence ${\mathcal {O}}_{X}^{m}|_{U}\to {\mathcal {O}}_{X}^{n}|_{U}\to {\mathcal {M}}|_{U}\to 0$ .) Paralleling the notion of acoherent module, ${\mathcal {M}}$ is called acoherent sheaf if it is of finite type and if, for every open set $U {\displaystyle U}$ and every morphism of sheaves $\phi :{\mathcal {O}}_{X}^{n}\to {\mathcal {M}}$ (not necessarily surjective), the kernel of $\phi$ is of finite type. ${\mathcal {O}}_{X}$ iscoherent if it is coherent as a module over itself. Like for modules, coherence is in general a strictly stronger condition than finite presentation. TheOka coherence theorem states that the sheaf of holomorphic functions on acomplex manifold is coherent.

The étalé space of a sheaf

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In the examples above it was noted that some sheaves occur naturally as sheaves of sections. In fact, all sheaves of sets can be represented as sheaves of sections of a topological space called theétalé space, from the French word étalé[etale], meaning roughly "spread out". If $F\in {\text{Sh}}(X)$ is a sheaf over $X {\displaystyle X}$ , then theétalé space (sometimes called theétale space) of $F {\displaystyle F}$ is a topological space $E {\displaystyle E}$ together with alocal homeomorphism $\pi :E\to X$ such that the sheaf of sections $\Gamma (\pi ,-)$ of $\pi$ is $F {\displaystyle F}$ . The space $E {\displaystyle E}$ is usually very strange, and even if the sheaf $F {\displaystyle F}$ arises from a natural topological situation, $E {\displaystyle E}$ may not have any clear topological interpretation. For example, if $F {\displaystyle F}$ is the sheaf of sections of a continuous function $f:Y\to X$ , then $E=Y$ if and only if $f {\displaystyle f}$ is alocal homeomorphism.

The étalé space $E {\displaystyle E}$ is constructed from the stalks of $F {\displaystyle F}$ over $X {\displaystyle X}$ . As a set, it is theirdisjoint union and $\pi$ is the obvious map that takes the value $x {\displaystyle x}$ on the stalk of $F {\displaystyle F}$ over $x\in X$ . The topology of $E {\displaystyle E}$ is defined as follows. For each element $s\in F(U)$ and each $x\in U$ , we get a germ of $s {\displaystyle s}$ at $x {\displaystyle x}$ , denoted $[s]_{x}$ or $s_{x}$ . These germs determine points of $E {\displaystyle E}$ . For any $U {\displaystyle U}$ and $s\in F(U)$ , the union of these points (for all $x\in U$ ) is declared to be open in $E {\displaystyle E}$ . Notice that each stalk has thediscrete topology assubspace topology. A morphism between two sheaves determine a continuous map of the corresponding étalé spaces that is compatible with the projection maps (in the sense that every germ is mapped to a germ over the same point). This makes the construction into a functor.

The construction above determines anequivalence of categories between the category of sheaves of sets on $X {\displaystyle X}$ and the category of étalé spaces over $X {\displaystyle X}$ . The construction of an étalé space can also be applied to a presheaf, in which case the sheaf of sections of the étalé space recovers the sheaf associated to the given presheaf.

This construction makes all sheaves intorepresentable functors on certain categories of topological spaces. As above, let $F {\displaystyle F}$ be a sheaf on $X {\displaystyle X}$ , let $E {\displaystyle E}$ be its étalé space, and let $\pi :E\to X$ be the natural projection. Consider theovercategory ${\text{Top}}/X$ of topological spaces over $X {\displaystyle X}$ , that is, thecategory of topological spaces together with fixed continuous maps to $X {\displaystyle X}$ . Every object of this category is a continuous map $f:Y\to X$ , and a morphism from $Y\to X$ to $Z\to X$ is a continuous map $Y\to Z$ that commutes with the two maps to $X {\displaystyle X}$ . There is a functor

$\Gamma :{\text{Top}}/X\to {\text{Sets}}$

sending an object $f:Y\to X$ to $f^{-1}F(Y)$ . For example, if $i:U\hookrightarrow X$ is the inclusion of an open subset, then

$\Gamma (i)=f^{-1}F(U)=F(U)=\Gamma (F,U)$

and for the inclusion of a point $i:\{x\}\hookrightarrow X$ , then

$\Gamma (i)=f^{-1}F(\{x\})=F|_{x}$

is the stalk of $F {\displaystyle F}$ at $x {\displaystyle x}$ . There is a natural isomorphism

$(f^{-1}F)(Y)\cong \operatorname {Hom} _{\mathbf {Top} /X}(f,\pi )$ ,

which shows that $\pi :E\to X$ (for the étalé space) represents the functor $\Gamma$ .

$E {\displaystyle E}$ is constructed so that the projection map $\pi$ is a covering map. In algebraic geometry, the natural analog of a covering map is called anétale morphism. Despite its similarity to "étalé", the word étale[etal] has a different meaning in French. It is possible to turn $E {\displaystyle E}$ into ascheme and $\pi$ into a morphism of schemes in such a way that $\pi$ retains the same universal property, but $\pi$ isnot in general an étale morphism because it is not quasi-finite. It is, however,formally étale.

The definition of sheaves by étalé spaces is older than the definition given earlier in the article. It is still common in some areas of mathematics such asmathematical analysis.

Sheaf cohomology

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Main article:Sheaf cohomology

In contexts where the open set $U {\displaystyle U}$ is fixed, and the sheaf is regarded as a variable, the set $F(U)$ is also often denoted $\Gamma (U,F).$

As was noted above, this functor does not preserve epimorphisms. Instead, an epimorphism of sheaves ${\mathcal {F}}\to {\mathcal {G}}$ is a map with the following property: for any section $g\in {\mathcal {G}}(U)$ there is a covering ${\mathcal {U}}=\{U_{i}\}_{i\in I}$ where

$U=\bigcup _{i\in I}U_{i}$

of open subsets, such that the restriction $g|_{U_{i}}$ are in the image of ${\mathcal {F}}(U_{i})$ . However, $g {\displaystyle g}$ itself need not be in the image of ${\mathcal {F}}(U)$ . A concrete example of this phenomenon is the exponential map

{\mathcal {O}}{\stackrel {\exp }{\to }}{\mathcal {O}}^{\times }

between the sheaf ofholomorphic functions and non-zero holomorphic functions. This map is an epimorphism, which amounts to saying that any non-zero holomorphic function $g {\displaystyle g}$ (on some open subset in $\mathbb {C}$ , say), admits acomplex logarithmlocally, i.e., after restricting $g {\displaystyle g}$ to appropriate open subsets. However, $g {\displaystyle g}$ need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for anexact sequence of sheaves of abelian groups

0\to {\mathcal {F}}_{1}\to {\mathcal {F}}_{2}\to {\mathcal {F}}_{3}\to 0,

(i.e., an epimorphism ${\mathcal {F}}_{2}\to {\mathcal {F}}_{3}$ whose kernel is ${\mathcal {F}}_{1}$ ), there is a long exact sequence $0\to \Gamma (U,{\mathcal {F}}_{1})\to \Gamma (U,{\mathcal {F}}_{2})\to \Gamma (U,{\mathcal {F}}_{3})\to H^{1}(U,{\mathcal {F}}_{1})\to H^{1}(U,{\mathcal {F}}_{2})\to H^{1}(U,{\mathcal {F}}_{3})\to H^{2}(U,{\mathcal {F}}_{1})\to \dots$ By means of this sequence, the first cohomology group $H^{1}(U,{\mathcal {F}}_{1})$ is a measure for the non-surjectivity of the map between sections of ${\mathcal {F}}_{2}$ and ${\mathcal {F}}_{3}$ .

There are several different ways of constructing sheaf cohomology.Grothendieck (1957) introduced them by defining sheaf cohomology as thederived functor of $\Gamma$ . This method is theoretically satisfactory, but, being based oninjective resolutions, of little use in concrete computations.Godement resolutions are another general, but practically inaccessible approach.

Computing sheaf cohomology

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Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions bysoft sheaves,fine sheaves, andflabby sheaves (also known asflasque sheaves from the Frenchflasque meaning flabby). For example, apartition of unity argument shows that the sheaf of smooth functions on a manifold is soft. The higher cohomology groups $H^{i}(U,{\mathcal {F}})$ for $i>0$ vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, thede Rham complex is a resolution of the constant sheaf ${\underline {\mathbf {R} }}$ on any smooth manifold, so the sheaf cohomology of ${\underline {\mathbf {R} }}$ is equal to itsde Rham cohomology.

A different approach is byČech cohomology. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing thecoherent sheaf cohomology of complex projective space $\mathbb {P} ^{n}$ .^[11] It relates sections on open subsets of the space to cohomology classes on the space. In most cases, Čech cohomology computes the same cohomology groups as the derived functor cohomology. However, for some pathological spaces, Čech cohomology will give the correct $H^{1}$ but incorrect higher cohomology groups. To get around this,Jean-Louis Verdier developedhypercoverings. Hypercoverings not only give the correct higher cohomology groups but also allow the open subsets mentioned above to be replaced by certain morphisms from another space. This flexibility is necessary in some applications, such as the construction ofPierre Deligne'smixed Hodge structures.

Many other coherent sheaf cohomology groups are found using an embedding $i:X\hookrightarrow Y$ of a space $X {\displaystyle X}$ into a space with known cohomology, such as $\mathbb {P} ^{n}$ , or someweighted projective space. In this way, the known sheaf cohomology groups on these ambient spaces can be related to the sheaves $i_{*}{\mathcal {F}}$ , giving $H^{i}(Y,i_{*}{\mathcal {F}})\cong H^{i}(X,{\mathcal {F}})$ . For example, computing thecoherent sheaf cohomology of projective plane curves is easily found. One big theorem in this space is theHodge decomposition found using aspectral sequence associated to sheaf cohomology groups, proved by Deligne.^[12]^[13] Essentially, the $E_{1}$ -page with terms

$E_{1}^{p,q}=H^{p}(X,\Omega _{X}^{q})$

the sheaf cohomology of asmooth projective variety $X {\displaystyle X}$ , degenerates, meaning $E_{1}=E_{\infty }$ . This gives the canonical Hodge structure on the cohomology groups $H^{k}(X,\mathbb {C} )$ . It was later found these cohomology groups can be easily explicitly computed usingGriffiths residues. SeeJacobian ideal. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties,the decomposition theorem, paving the path forMixed Hodge modules.

Another clean approach to the computation of some cohomology groups is theBorel–Bott–Weil theorem, which identifies the cohomology groups of someline bundles onflag manifolds withirreducible representations ofLie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space andgrassmann manifolds.

In many cases there is a duality theory for sheaves that generalizesPoincaré duality. SeeGrothendieck duality andVerdier duality.

Derived categories of sheaves

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Thederived category of the category of sheaves of, say, abelian groups on some spaceX, denoted here as $D(X)$ , is the conceptual haven for sheaf cohomology, by virtue of the following relation:

H^{n}(X,{\mathcal {F}})=\operatorname {Hom} _{D(X)}(\mathbf {Z} ,{\mathcal {F}}[n]).

The adjunction between $f^{-1}$ , which is the left adjoint of $f_{*}$ (already on the level of sheaves of abelian groups) gives rise to an adjunction

f^{-1}:D(Y)\rightleftarrows D(X):Rf_{*}

(for

f:X\to Y

where $Rf_{*}$ is the derived functor. This latter functor encompasses the notion of sheaf cohomology since $H^{n}(X,{\mathcal {F}})=R^{n}f_{*}{\mathcal {F}}$ for $f:X\to \{*\}$ .

Image functors for sheaves
direct image $f_{*}$
inverse image $f^{*}$
direct image with compact support $f_{!}$
exceptional inverse image $Rf^{!}$
$f^{}\leftrightarrows f_{}$
$(R)f_{!}\leftrightarrows (R)f^{!}$
Base change theorems
v t e

Like $f_{*}$ , the direct image with compact support $f_{!}$ can also be derived. By virtue of the following isomorphism $Rf_{!}{\mathcal {F}}$ parametrizes thecohomology with compact support of thefibers of $f {\displaystyle f}$ :

(R^{i}f_{!}{\mathcal {F}})_{y}=H_{c}^{i}(f^{-1}(y),{\mathcal {F}}).

^[14]

This isomorphism is an example of abase change theorem. There is another adjunction

Rf_{!}:D(X)\rightleftarrows D(Y):f^{!}.

Unlike all the functors considered above, the twisted (or exceptional) inverse image functor $f^{!}$ is in general only defined on the level ofderived categories, i.e., the functor is not obtained as the derived functor of some functor betweenabelian categories. If $f:X\to \{*\}$ andX is a smoothorientable manifold of dimensionn, then

f^{!}{\underline {\mathbf {R} }}\cong {\underline {\mathbf {R} }}[n].

^[15]

This computation, and the compatibility of the functors with duality (seeVerdier duality) can be used to obtain a high-brow explanation ofPoincaré duality. In the context of quasi-coherent sheaves on schemes, there is a similar duality known ascoherent duality.

Perverse sheaves are certain objects in $D(X)$ , i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry ofsingularities.^[16]

Derived categories of coherent sheaves and the Grothendieck group

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Another important application of derived categories of sheaves is with the derived category ofcoherent sheaves on a scheme $X {\displaystyle X}$ denoted $D_{Coh}(X)$ . This was used by Grothendieck in his development ofintersection theory^[17] usingderived categories andK-theory, that the intersection product of subschemes $Y_{1},Y_{2}$ is represented inK-theory as

$[Y_{1}]\cdot [Y_{2}]=[{\mathcal {O}}_{Y_{1}}\otimes _{{\mathcal {O}}_{X}}^{\mathbf {L} }{\mathcal {O}}_{Y_{2}}]\in K({\text{Coh(X)}})$

where ${\mathcal {O}}_{Y_{i}}$ arecoherent sheaves defined by the ${\mathcal {O}}_{X}$ -modules given by theirstructure sheaves.

Sites and topoi

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Main articles:Grothendieck topology andTopos

André Weil'sWeil conjectures stated that there was acohomology theory foralgebraic varieties overfinite fields that would give an analogue of theRiemann hypothesis. The cohomology of a complex manifold can be defined as the sheaf cohomology of the locally constant sheaf ${\underline {\mathbf {C} }}$ in the Euclidean topology, which suggests defining a Weil cohomology theory in positive characteristic as the sheaf cohomology of a constant sheaf. But the only classical topology on such a variety is theZariski topology, and the Zariski topology has very few open sets, so few that the cohomology of any Zariski-constant sheaf on an irreducible variety vanishes (except in degree zero).Alexandre Grothendieck solved this problem by introducingGrothendieck topologies, which axiomatize the notion ofcovering. Grothendieck's insight was that the definition of a sheaf depends only on the open sets of a topological space, not on the individual points. Once he had axiomatized the notion of covering, open sets could be replaced by other objects. A presheaf takes each one of these objects to data, just as before, and a sheaf is a presheaf that satisfies the gluing axiom with respect to our new notion of covering. This allowed Grothendieck to defineétale cohomology andℓ-adic cohomology, which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called asite. A category of sheaves on a site is called atopos or aGrothendieck topos. The notion of a topos was later abstracted byWilliam Lawvere and Miles Tierney to define anelementary topos, which has connections tomathematical logic.

History

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The first origins ofsheaf theory are hard to pin down – they may be co-extensive with the idea ofanalytic continuation^{[clarification needed]}. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work oncohomology.

1936Eduard Čech introduces thenerve construction, for associating asimplicial complex to an open covering.
1938Hassler Whitney gives a 'modern' definition of cohomology, summarizing the work sinceJ. W. Alexander andKolmogorov first definedcochains.
1943Norman Steenrod publishes on homologywithlocal coefficients.^[18]
1945Jean Leray publishes work carried out as aprisoner of war, motivated by provingfixed-point theorems for application toPDE theory; it is the start of sheaf theory andspectral sequences.^[19]
1947Henri Cartan reproves thede Rham theorem by sheaf methods, in correspondence withAndré Weil (seeDe Rham–Weil theorem). Leray gives a sheaf definition in his courses via closed sets (the latercarapaces).
1948 The Cartan seminar writes up sheaf theory for the first time.
1950 The "second edition" sheaf theory from the Cartan seminar: thesheaf space (espace étalé) definition is used, with stalkwise structure.Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same timeKiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, inseveral complex variables.
1951 The Cartan seminar provestheorems A and B, based on Oka's work.
1953 The finiteness theorem forcoherent sheaves in the analytic theory is proved by Cartan andJean-Pierre Serre,^[20] as isSerre duality.
1954 Serre's paperFaisceaux algébriques cohérents^[21] (published in 1955) introduces sheaves intoalgebraic geometry. These ideas are immediately exploited byFriedrich Hirzebruch, who writes a major 1956 book on topological methods.
1955Alexander Grothendieck in lectures inKansas definesabelian category andpresheaf, and by usinginjective resolutions allows direct use of sheaf cohomology on all topological spaces, asderived functors.
1956Oscar Zariski's reportAlgebraic sheaf theory^[22]
1957 Grothendieck'sTohoku paper^[23] rewriteshomological algebra; he provesGrothendieck duality (i.e., Serre duality for possiblysingular algebraic varieties).
1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing:schemes and general sheaves on them,local cohomology,derived categories (with Verdier), andGrothendieck topologies. There emerges also his influential schematic idea of 'six operations' in homological algebra.
1958Roger Godement's book on sheaf theory is published. At around this timeMikio Sato proposes hishyperfunctions, which will turn out to have sheaf-theoretic nature.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted toalgebraic topology. It was later discovered that the logic in categories of sheaves isintuitionistic logic (this observation is now often referred to asKripke–Joyal semantics, but probably should be attributed to a number of authors).

Notes

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^Eisenbud, David; Harris, Joe (6 April 2006),The Geometry of Schemes,GTM, New York, NY: Springer, pp. 11–18,ISBN 978-0-387-22639-2
^Tennison, B. R. (1975),Sheaf theory,Cambridge University Press,MR 0404390
^Bredon (1997, Chapter V, §1)
^Demailly, Jean-Pierre."Complex Analytic and Differential Geometry"(PDF).Archived(PDF) from the original on 28 August 2020.
^Cartan, Henri."Variétés analytiques complexes et cohomologie"(PDF).Archived(PDF) from the original on 8 October 2020.
^^a ^b"differential geometry - Holomorphic functions on a complex compact manifold are only constants".Mathematics Stack Exchange. Retrieved2020-10-07.
^Hawley, Newton S. (1950). "A Theorem on Compact Complex Manifolds".Annals of Mathematics.52 (3):637–641.doi:10.2307/1969438.JSTOR 1969438.
^SGA 4 II 3.0.5
^Iversen (1986, Chapter VII)
^Ramanan (2005)
^Hartshorne (1977), Theorem III.5.1.
^Deligne, Pierre (1971)."Théorie de Hodge : II".Publications Mathématiques de l'IHÉS.40:5–57.doi:10.1007/BF02684692.S2CID 118967613.
^Deligne, Pierre (1974)."Théorie de Hodge : III".Publications Mathématiques de l'IHÉS.44:5–77.doi:10.1007/BF02685881.S2CID 189777706.
^Iversen (1986, Chapter VII, Theorem 1.4)
^Kashiwara & Schapira (1994, Chapter III, §3.1)
^de Cataldo & Migliorini (2010)
^Grothendieck."Formalisme des intersections sur les schema algebriques propres".
^Steenrod, N. E. (1943). "Homology with Local Coefficients".Annals of Mathematics.44 (4):610–627.doi:10.2307/1969099.JSTOR 1969099.
^Dieudonné, Jean (1989).A history of algebraic and differential topology 1900–1960. Birkhäuser. pp. 123–141.ISBN 978-0-8176-3388-2.
^Cartan, Henri; Serre, Jean-Pierre (1953)."Un théorème de finitude concernant les variétés analytiques compactes".Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris.237:128–130.Zbl 0050.17701.
^Serre, Jean-Pierre (1955),"Faisceaux algébriques cohérents"(PDF),Annals of Mathematics, Second Series,61 (2):197–278,doi:10.2307/1969915,ISSN 0003-486X,JSTOR 1969915,MR 0068874
^Zariski, Oscar (1956), "Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory",Bulletin of the American Mathematical Society,62 (2):117–141,doi:10.1090/S0002-9904-1956-10018-9,ISSN 0002-9904
^Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique",The Tohoku Mathematical Journal, Second Series,9 (2):119–221,doi:10.2748/tmj/1178244839,ISSN 0040-8735,MR 0102537

References

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Bredon, Glen E. (1997),Sheaf theory, Graduate Texts in Mathematics, vol. 170 (2nd ed.),Springer-Verlag,ISBN 978-0-387-94905-5,MR 1481706 (oriented towards conventional topological applications)
de Cataldo, Andrea Mark; Migliorini, Luca (2010)."What is a perverse sheaf?"(PDF).Notices of the American Mathematical Society.57 (5):632–4.arXiv:1004.2983.Bibcode:2010arXiv1004.2983D.MR 2664042.
Godement, Roger (2006) [1973],Topologie algébrique et théorie des faisceaux, Paris: Hermann,ISBN 2705612521,MR 0345092
Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique",The Tohoku Mathematical Journal, Second Series,9 (2):119–221,doi:10.2748/tmj/1178244839,ISSN 0040-8735,MR 0102537
Hirzebruch, Friedrich (1995),Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag,ISBN 978-3-540-58663-0,MR 1335917 (updated edition of a classic using enough sheaf theory to show its power)
Iversen, Birger (1986),Cohomology of sheaves, Universitext, Springer,doi:10.1007/978-3-642-82783-9,ISBN 3-540-16389-1,MR 0842190
Kashiwara, Masaki; Schapira, Pierre (1994),Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag,ISBN 978-3-540-51861-7,MR 1299726 (advanced techniques such as thederived category andvanishing cycles on the most reasonable spaces)
Mac Lane, Saunders;Moerdijk, Ieke (1994),Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext, Springer-Verlag,ISBN 978-0-387-97710-2,MR 1300636 (category theory and toposes emphasised)
Martin, William T.;Chern, Shiing-Shen;Zariski, Oscar (1956), "Scientific report on the Second Summer Institute, several complex variables",Bulletin of the American Mathematical Society,62 (2):79–141,doi:10.1090/S0002-9904-1956-10013-X,ISSN 0002-9904,MR 0077995
Ramanan, S. (2005),Global calculus, Graduate Studies in Mathematics, vol. 65, American Mathematical Society,doi:10.1090/gsm/065,ISBN 0-8218-3702-8,MR 2104612
Seebach, J. Arthur; Seebach, Linda A.;Steen, Lynn A. (1970), "What is a Sheaf",American Mathematical Monthly,77 (7):681–703,doi:10.1080/00029890.1970.11992563,MR 0263073,S2CID 203043621
Serre, Jean-Pierre (1955),"Faisceaux algébriques cohérents"(PDF),Annals of Mathematics, Second Series,61 (2):197–278,doi:10.2307/1969915,ISSN 0003-486X,JSTOR 1969915,MR 0068874
Swan, Richard G. (1964),The Theory of Sheaves, Chicago lectures in mathematics (3 ed.),University of Chicago Press,ISBN 9780226783291{{citation}}:ISBN / Date incompatibility (help) (concise lecture notes)
Tennison, Barry R. (1975),Sheaf theory, London Mathematical Society Lecture Note Series, vol. 20,Cambridge University Press,ISBN 978-0-521-20784-3,MR 0404390 (pedagogic treatment)
Rosiak, Daniel (2022).Sheaf theory through examples. Cambridge, Massachusetts.doi:10.7551/mitpress/12581.001.0001.ISBN 978-0-262-37042-4.OCLC 1333708310.S2CID 253133215.{{cite book}}: CS1 maint: location missing publisher (link) (introductory book with open access)