** Sheaves **

Today, I explain what means.

**1. Basics **

** 1.1. Presheaves **

Definition 1topological space. A presheaf of -modules on is a mapfrom open subsets of to -modules, such that, if , there is a restrction homomorphism such that– ,

– when .

A presheaf is a sheaf if whenever are given and , there exists whose restriction to is .

Example 1The constant presheaf is not a sheaf in general. By definition, the constant sheaf is the sheaf whose sections are locally constant functions.Given a closed subset , is defined by locally constant functions on .

Given an open subset , is defined by locally constant functions on with closed support in .

One defines morphisms of presheaves, and direct images.

** 1.2. Localization and sheafification **

The *stalk* of a presheaf at a point is the direct limit of as varies in the direct system of neighborhoods of . It is natural under morphisms.

Example 2The stalk at of the sheaf of continuous functions is the space of germs of continuous functions, by definition.Assume that is ocally connected. Given a closed subset , is and if .

Given an open subset , is and if .

Definition 2Let be a presheaf. The sheafification of is universal for sheaves with a morphism from inducing an isomorphism on stalks.

**Inverse image sheaf**. Given a map and a sheaf on , is a presheaf on . Its sheafification is denoted by .

Example 3Let be the injection of a closed subset. Then the direct image is , the inverse image .

**Shriek image sheaf**. Let be a sheaf on . Define

Example 4Let be the injection of an open subset. Then .

**2. Cohomology **

** 2.1. Cech cohomology **

I start with a more geometric notion : \v Cech cohomology.

Fix an open covering of . A -cochain is a collection of sections of , one for each -fold intersection. The coboundary operator is

One checks that . The cohomology of is .

If is a refinement of , restrictions induce a map . The direct limit, where varies over the directed set of open coverings of , is the *\v Cech cohomology* .

Theorem 3 (Leray acyclic covering theorem)If is acyclic, then .

Acyclic means that all intersections have vanishing cohomology.

** 2.2. Injective resolutions **

Sheaves and presheaves on form abelian categories. Indeed, one can define kernels of morphisms, and these are sheaves and presheaves. The cokernel needs be sheafified.

Lemma 4A sequence of sheaves is exact if and only if the corresponding sequences of stalks are exact at each point.

The “space of sections” functor -modules is left exact. This allows an alternative definition of cohomology, folowing general principles.

In a category, a monomorphism is a morphism such that for all , , .

An object in a category is *injective* if for every monomorphism , any morphism factors through .

Definition 5A category has enough injectives if every object has a monomorphism into an injective object.

Theorem 6The category of sheaves on has enough injectives.

Proposition 7In a category with enough injectives, for every object , there exists an exact sequence

with injective ,… It is called an injective resolution of .

No uniqueness, but any two such resolutions are linked by maps.

Definition 8Let be a left exact functor. Let be an object. Define to be the -th cohomology group of the complexwhere

is a injection resolution of . It turns out not to depend on the choice of injective resolution.

** 2.3. Acyclic resolutions **

Say an object is -acyclic if for all . Of course, injectives are acyclic. Such objects are easier to find, and they turn out to suffice to compute cohomology.

Theorem 9Letbe a resolution of with -acyclic ‘s. Then the complex is quasi-isomorphic to

In particular, is the -th cohomology group of

Say a sheaf is *flabby* if every section on some open set can be extended to the whole space.

Say a sheaf is *soft* if for every closed subset , every section defined in some neighborhood can be extended to the whole space after restriction to a smaller neighborhood.

Example 5Continuous functions form a soft, non flabby, sheaf.

Flabby sheaves and soft sheaves are -acyclic objects.

**De Rham cohomology**. Let be a manifold. Differential forms form soft sheaves. Therefore the resolution of the constant sheaf by differential forms can be used to compute the cohomology of the constant sheaf .

**Singular cohomology**. Singular cochains form a flabby sheaf. Therefore the resolution of the constant sheaf by singular cochains can be used to compute the cohomology of the constant sheaf .

Given a sheaf and an open covering , spaces of cochains form sheaves. The coboundary produces a resolution of . If is acyclic, this is an acyclic resolution. Therefore \v Ceck cohomology equals cohomology.

Later, we shall see that cohomology is related with the derived functor of the functor.

17 janvier : microsupport.