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Wikipedia / Sheaf (mathematics)

In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as setss, abelian groups, ringss) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous 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 sets covering the original open set (intuitively, every datum is the sum of its constituent data).

The field of mathematics that studies sheaves is called sheaf theory.

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

There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, 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 in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts, several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. Especially in algebraic geometry and the theory of complex manifolds, sheaf cohomology provides a powerful link between topological and geometric properties of spaces. Sheaves also provide the basis for the theory of D-moduless, which provide applications to the theory of differential 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 some Grothendieck topology, have provided applications to mathematical logic and to number theory. == Definitions and examples ==

In many mathematical branches, several structures defined on a topological space (e.g., a differentiable manifold) can be naturally localised or restricted to open subsets: typical examples include continuous real-valued or complex-times differentiable (real-valued or complex-valued) functions, bounded real-valued functions, vector fields, and sections of any vector 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

be a topological space. A of sets consists of the following data:

  • . The elements in this set are called the sections are called the global sections.

  • are called restriction morphisms by analogy with restriction of functions.

The restriction morphisms are required to satisfy two additional (functorial) properties:

. A concise functorial reformulation of this definition is given further below.

.

. This presheaf is called the constant presheaf.

Sheaves

. A sheaf is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.

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

  1. (Locality.

  2. (Gluing.

.

whose existence is guaranteed by axiom 2 is called the gluing, concatenation, or collation satisfying the agreement precondition of axiom 2 are often called compatible ; thus axioms 1 and 2 together state that any collection of pairwise compatible sections can be uniquely glued together. A separated presheaf, or monopresheaf, is a presheaf satisfying axiom 1.

. By contrast, the constant presheaf is usually not a sheaf as it fails to satisfy the locality axiom on the empty set (this is explained in more detail at constant sheaf).

being particularly common, presumably for the French word for sheaf, faisceau is also common.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a basis 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, namely quasi-coherent sheaves. Here the topological space in question is the R, whose points are the prime ideals form a basis for the Zariski topology is the localization.

is a sheaf if and only if are abelian sheaves is a sheaf, since projective limits commute 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 commute with inductive limits.

Further examples

Sheaf of sections of a continuous map

by setting

is commonly called a section is the projection of a fiber bundle onto its base space. For example, the sheaves of smooth functions are the sheaves of sections of the trivial bundle.

Another example: the sheaf of sections of

the set of branches of the complex logarithm.

, the trivial group, or the zero map otherwise.

Sheaves on manifolds

, this sheaf is called the structure sheaf. Differential forms. In all these examples, the restriction morphisms are given by restricting functions or forms.

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 a cosheaf, a dual concept where the restriction maps go in the opposite direction than with sheaves. However, taking the dual of these vector spaces does give a sheaf, the sheaf of distributions.

Presheaves that are not sheaves

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

Motivating sheaves from complex analytic spaces and algebraic geometry

One of the historical motivations for sheaves have come from studying complex manifolds, complex analytic geometry, and scheme theory from algebraic geometry 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

One of the main historical motivations for introducing sheaves was constructing a device which keeps track of holomorphic functions on complex manifolds. For example, on a compact (like complex projective space or the vanishing locus in projective space of a homogeneous polynomial), the only holomorphic functions

are the constant functions., are isomorphic. Contrast this with smooth manifolds, of which there exist plenty.

when we want to emphasize the space the structure sheaf is associated to.

Tracking submanifolds with sheaves

. This kind of formalism was found to be extremely powerful and motivates a lot of homological algebra such as sheaf cohomology since an intersection theory can be built using these kinds of sheaves from the Serre intersection formula.

Operations with sheaves

Morphisms

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.

. A morphism, the following diagram is commutative.

,

.

form a category. The general categorical notions of mono-, epi- and isomorphisms can therefore be applied to sheaves.

is a full subcategory of the category of presheaves being a natural transformation between the two sheaves seen as functors.

are not always surjective for epimorphisms of sheaves is equivalent to non-exactness of the global sections functor—or equivalently, to non-triviality of sheaf cohomology.

Stalks of a sheaf

The stalk, generalizing the germs of functions. Here, "around" means that, conceptually speaking, one looks at smaller and smaller neighborhoods 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

the direct limit, and two such sections are considered equivalent if their restrictions agree on a smaller neighborhood.

to its germ. This generalises the usual definition of a germ.

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, the global information present in a sheaf, i.e., the global sections, typically carry less information. For example, for a compact is constant by Liouville's theorem.

Turning a presheaf into a sheaf

called the sheafification or sheaf associated to the presheaf. For example, the sheafification of the constant presheaf (see above) is called the constant sheaf. Despite its name, its sections are locally constant functions.

can be constructed using the étalé space, namely as the sheaf of sections of the map

.

by a sheaf is made precise using the following universal property is the left adjoint functor to the inclusion functor (or forgetful functor is the unit of the adjunction. In this way, the category of sheaves turns into a Giraud 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

is a subsheaf of abelian groups, then the quotient sheaf; in other words, the quotient sheaf fits into an exact sequence of sheaves of abelian groups;

(this is also called a sheaf extension.)

). The sheaf hom, denoted by,

.

All of these operations extend to sheaves of modules over a sheaf of rings is the constant sheaf.

Basic functoriality

and vice versa.

Direct image

The pushforward (also known as direct image is the sheaf defined by

mentioned above:

is regarded as a sheaf on the singleton.

For a map between locally compact spaces, the direct image with compact support is a subsheaf of the direct image. whose support is mapped properly, but in general they disagree.

Inverse image

The pullback or inverse image) is called locally constant, such sheaves are equivalent to representations of the fundamental group.

is more involved; it is detailed at inverse image functor is as above:

.

Extension by zero

of an open subset, the extension by zero (pronounced "j lower shriek is the sheafification of the presheaf defined by

:

is a locally closed subset be the natural inclusions. Then the extension by zero.

to ones on the strata of a stratification into smaller, locally closed subsets.

Complements

Sheaves in more general categories

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 real vector space, and restriction is a linear map between these vector spaces.

to be the posetal category is the same as a contravariant functor. Morphisms in this category of functors, also known as natural transformations, are the same as the morphisms defined above, as can be seen by unraveling the definitions.

admits all limits-valued presheaf is a sheaf if the following diagram is an equalizer:

Here the first map is the product of the restriction maps

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

and

is an abelian category, this condition can also be rephrased by requiring that there is an exact sequence

to be the terminal object.

Ringed spaces and sheaves of modules

In several geometrical disciplines, including algebraic geometry and differential geometry is called a ringed space of the structure sheaf are local rings, in which case the pair is called a locally ringed space.

. The property of being a locally. Some authors actually define (respectively holomorphic) functions. Similarly, schemess, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring.

Given a ringed space, a sheaf of modules.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between vector bundles and locally free sheaves consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are D-moduless, that is, modules over the sheaf of differential operators are the same as sheaves of abelian groups in the sense above.

. See inverse image functor.

Finiteness conditions for sheaves of modules

Finiteness conditions for module over commutative rings is called finitely generated (respectively finitely presented.) Paralleling the notion of a coherent module is called a coherent sheaf is coherent if it is coherent as a module over itself. Like for modules, coherence is in general a strictly stronger condition than finite presentation. The Oka coherence theorem states that the sheaf of holomorphic functions on a complex manifold is coherent.

The étalé space of a sheaf

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 the étalé space, from the French word , then the étalé space (sometimes called the étale space together with a local homeomorphism is a local homeomorphism.

. As a set, it is their disjoint union. Notice that each stalk has the discrete topology as subspace 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 an equivalence of categories. 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 into representable functors be the natural projection. Consider the overcategory, that is, the category of topological spaces. There is a functor is the inclusion of an open subset, then, then. There is a natural isomorphism,.

is a covering map. In algebraic geometry, the natural analog of a covering map is called an étale morphism. Despite its similarity to "étalé", the word étale into a scheme is not 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 as mathematical analysis.

Sheaf cohomology

where between the sheaf of holomorphic functions, say), admits a complex logarithm locally need not have a logarithm globally.

Sheaf cohomology captures this phenomenon. More precisely, for an exact sequence.

There are several different ways of constructing sheaf cohomology. introduced them by defining sheaf cohomology as the derived functor. This method is theoretically satisfactory, but, being based on injective resolutions, of little use in concrete computations. Godement resolutions are another general, but practically inaccessible approach.

Computing sheaf cohomology

Especially in the context of sheaves on manifolds, sheaf cohomology can often be computed using resolutions by soft sheaves, fine sheaves, and flabby sheaves (also known as flasque sheaves from the French flasque meaning flabby). For example, a partition of unity vanish for soft sheaves, which gives a way of computing cohomology of other sheaves. For example, the de Rham complex is equal to its de Rham cohomology.

A different approach is by Čech cohomology. Čech cohomology was the first cohomology theory developed for sheaves and it is well-suited to concrete calculations, such as computing the coherent sheaf cohomology. but incorrect higher cohomology groups. To get around this, Jean-Louis Verdier developed hypercoverings. 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 of Pierre Deligne's mixed Hodge structures.

, or some weighted projective space. For example, computing the coherent sheaf cohomology of projective plane curves is easily found. One big theorem in this space is the Hodge decomposition found using a spectral sequence associated to sheaf cohomology groups, proved by Deligne.-page with termsthe sheaf cohomology of a smooth projective variety. It was later found these cohomology groups can be easily explicitly computed using Griffiths residues. See Jacobian ideal. These kinds of theorems lead to one of the deepest theorems about the cohomology of algebraic varieties, the decomposition theorem, paving the path for Mixed Hodge modules.

Another clean approach to the computation of some cohomology groups is the Borel–Bott–Weil theorem, which identifies the cohomology groups of some line bundles on flag manifolds with irreducible representations of Lie groups. This theorem can be used, for example, to easily compute the cohomology groups of all line bundles on projective space and grassmann manifolds.

In many cases there is a duality theory for sheaves that generalizes Poincaré duality. See Grothendieck duality and Verdier duality.

Derived categories of sheaves

The derived category of the category of sheaves of, say, abelian groups on some space X.

parametrizes the cohomology with compact support of the fibers: This isomorphism is an example of a base change theorem is in general only defined on the level of derived categories, i.e., the functor is not obtained as the derived functor of some functor between abelian categories and X is a smooth orientable manifold of dimension n, then This computation, and the compatibility of the functors with duality (see Verdier duality) can be used to obtain a high-brow explanation of Poincaré duality. In the context of quasi-coherent sheaves on schemes, there is a similar duality known as coherent duality.

Perverse sheaves, i.e., complexes of sheaves (but not in general sheaves proper). They are an important tool to study the geometry of singularities.

Derived categories of coherent sheaves and the Grothendieck group

Another important application of derived categories of sheaves is with the derived category of coherent sheaves. This was used by Grothendieck in his development of intersection theory using derived categories and K-theory is represented in K-theory as

are coherent sheaves-modules given by their structure sheaves.

Sites and topoi

André Weil's Weil conjectures stated that there was a cohomology theory for algebraic varieties over finite fields that would give an analogue of the Riemann hypothesis 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 the Zariski 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 introducing Grothendieck topologies, which axiomatize the notion of covering. 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 define étale cohomology and ℓ-adic cohomology, which eventually were used to prove the Weil conjectures.

A category with a Grothendieck topology is called a site. A category of sheaves on a site is called a topos or a Grothendieck topos. The notion of a topos was later abstracted by William Lawvere and Miles Tierney to define an elementary topos, which has connections to mathematical logic.

History

The first origins of sheaf theory are hard to pin down; they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

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

See also

Notes

References

  • (oriented towards conventional topological applications)

  • (Updated edition of a classic using enough sheaf theory to show its power.)

  • (Advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces.)

  • (Category theory and toposes emphasised.)

  • (Concise lecture notes.)

  • (pedagogic treatment)

  • (Introductory book with open access.)