Today, I explain what means.
Definition 1 topological 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 1 The 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 2 The 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 2 Let 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 3 Let 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 4 Let be the injection of an open subset. Then .
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 4 A 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 5 A category has enough injectives if every object has a monomorphism into an injective object.
Theorem 6 The category of sheaves on has enough injectives.
Proposition 7 In 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 8 Let be a left exact functor. Let be an object. Define to be the -th cohomology group of the complex
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 9 Let
be 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 5 Continuous 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.