Notes of Anne Vaugon’s lecture

Sheaves

Today, I explain what {R\Gamma(X,\mathcal{F})} means.

1. Basics

1.1. Presheaves

Definition 1 {X} topological space. A presheaf of {k}-modules on {X} is a mapfrom open subsets of {X} to {k}-modules, such that, if {V\subseteq U}, there is a restrction homomorphism {r_{VU}:\mathcal{F}(U)\rightarrow\mathcal{F}(V)} such that

{r_{UU}=Id_U},

{r_{WV}\circ r_{VU}=r_{WU}} when {W\subseteq V\subseteq U}.

A presheaf is a sheaf if whenever {s_\in\mathcal{F}(U_i)} are given and {r_{U_i \cap U_j,U_i}(s_i)=r_{U_i \cap U_j,U_j}(s_j)}, there exists {s\in\mathcal{F}(\bigcup_i U_i)} whose restriction to {U_i} is {s_i}.

Example 1 The constant presheaf is not a sheaf in general. By definition, the constant sheaf {k} is the sheaf whose sections are locally constant functions.

Given a closed subset {A\subset X}, {k_A} is defined by {k_A(U)=k(U\cap A)=\{}locally constant functions on {U\cap A\}}.

Given an open subset {\Omega\subset X}, {k_\Omega} is defined by {k_\Omega(U)=\{}locally constant functions on {U\cap\Omega} with closed support in {U\}}.

One defines morphisms of presheaves, and direct images.

1.2. Localization and sheafification

The stalk {\mathcal{F}_x} of a presheaf {\mathcal{F}} at a point {x} is the direct limit of {\mathcal{F}(U)} as {U} varies in the direct system of neighborhoods of {x}. It is natural under morphisms.

Example 2 The stalk at {x} of the sheaf of continuous functions is the space of germs of continuous functions, by definition.

Assume that {X} is ocally connected. Given a closed subset {A\subset X}, {(k_A)_x=k} is {x\in A} and {0} if {x\notin A}.

Given an open subset {\Omega\subset X}, {(k_\Omega)_x=k} is {x\in \Omega} and {0} if {x\notin \Omega}.

Definition 2 Let {\mathcal{F}} be a presheaf. The sheafification {\tilde{\mathcal{F}}} of {\mathcal{F}} is universal for sheaves with a morphism from {\mathcal{F}} inducing an isomorphism on stalks.

Inverse image sheaf. Given a map {f:X\rightarrow Y} and a sheaf {\mathcal{G}} on {Y}, {f^{-1}\mathcal{G}} is a presheaf on {X}. Its sheafification is denoted by {Pf^{-1}\mathcal{G}}.

Example 3 Let {i:A\rightarrow X} be the injection of a closed subset. Then the direct image is {i_* k=k_A}, the inverse image {i^{-1}k_A=k}.

Shriek image sheaf. Let {\mathcal{F}} be a sheaf on {X}. Define

\displaystyle  \begin{array}{rcl}  f_{!}\mathcal{F}(U)=\{s\in\mathcal{F}(f^{-1}(U))\,;\,f\textrm{ is proper on the support of }s\}. \end{array}

Example 4 Let {i:\Omega\rightarrow X} be the injection of an open subset. Then {i_! k=k_\Omega}.

2. Cohomology

2.1. Cech cohomology

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

Fix an open covering {\mathcal{U}=\{U_i\}} of {X}. A {p}-cochain is a collection of sections of {\mathcal{F}}, one for each {p+1}-fold intersection. The coboundary operator is

\displaystyle  \begin{array}{rcl}  (\delta s)(i_0,\ldots,i_p)=\sum_{j=0}^{p+1}(-1)^{j+1}s(i_0,\ldots,\widehat{i_j},\ldots,i_{p+1}). \end{array}

One checks that {\delta\circ\delta=0}. The cohomology of {\delta} is {H^{\cdot} (\mathcal{U},\mathcal{F})}.

If {\mathcal{U}'} is a refinement of {\mathcal{U}}, restrictions induce a map {H^{\cdot} (\mathcal{U},\mathcal{F})\rightarrow H^{\cdot} (\mathcal{U}',\mathcal{F})}. The direct limit, where {\mathcal{U}} varies over the directed set of open coverings of {X}, is the \v Cech cohomology {H^{\cdot}(X,\mathcal{F})}.

Theorem 3 (Leray acyclic covering theorem) If {\mathcal{U}} is acyclic, then {H^{\cdot}(X,\mathcal{F})=H^{\cdot}(\mathcal{U},\mathcal{F})}.

Acyclic means that all intersections have vanishing cohomology.

2.2. Injective resolutions

Sheaves and presheaves on {X} 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 {\mathcal{F}\rightarrow\mathcal{G}\rightarrow\mathcal{H}} of sheaves is exact if and only if the corresponding sequences of stalks are exact at each point.

The “space of sections” functor {\Gamma_X:Sheaf(X)\rightarrow k}-modules is left exact. This allows an alternative definition of cohomology, folowing general principles.

In a category, a monomorphism is a morphism {f\in Mor(A,B)} such that for all {g_1}, {g_2 \in Mor(A,B)}, {f\circ g_1=f\circ g_2}.

An object {A} in a category is injective if for every monomorphism {f:A\rightarrow B}, any morphism {A\rightarrow I} factors through {f}.

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

Theorem 6 The category of sheaves on {X} has enough injectives.

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

\displaystyle  \begin{array}{rcl}  0\rightarrow B\rightarrow J_0 \rightarrow J_1\rightarrow \cdots \end{array}

with injective {J_0},… It is called an injective resolution of {B}.

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

Definition 8 Let {F} be a left exact functor. Let {A} be an object. Define {R^j F(A)} to be the {j}-th cohomology group of the complex

\displaystyle  \begin{array}{rcl}  0\rightarrow F(J_0) \rightarrow F(J_1)\rightarrow \cdots, \end{array}

where

\displaystyle  \begin{array}{rcl}  0\rightarrow B\rightarrow J_0 \rightarrow J_1\rightarrow \cdots \end{array}

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

2.3. Acyclic resolutions

Say an object {L} is {F}-acyclic if {R^j F(L)=0} for all {j\geq 1}. Of course, injectives are acyclic. Such objects are easier to find, and they turn out to suffice to compute cohomology.

Theorem 9 Let

\displaystyle  \begin{array}{rcl}  0\rightarrow A\rightarrow L_0 \rightarrow L_1\rightarrow \cdots \end{array}

be a resolution of {A} with {F}-acyclic {L_i}‘s. Then the complex {RF(A)} is quasi-isomorphic to

\displaystyle  \begin{array}{rcl}  0\rightarrow F(L_0) \rightarrow F(L_1)\rightarrow \cdots. \end{array}

In particular, {R^j F(A)} is the {j}-th cohomology group of

\displaystyle  \begin{array}{rcl}  0\rightarrow F(L_0) \rightarrow F(L_1)\rightarrow \cdots. \end{array}

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 {A}, 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 {\Gamma_X}-acyclic objects.

De Rham cohomology. Let {X} be a manifold. Differential forms form soft sheaves. Therefore the resolution of the constant sheaf {{\mathbb R}} by differential forms can be used to compute the cohomology of the constant sheaf {{\mathbb R}}.

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 {k}.

Given a sheaf {\mathcal{F}} and an open covering {\mathcal{U}}, spaces of cochains {C^p(\mathcal{U},\mathcal{F})} form sheaves. The coboundary produces a resolution of {\mathcal{F}}. If {\mathcal{U}} 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 {\Gamma_X} functor.

17 janvier : microsupport.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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