## 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

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.