Chapter 8

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 11 June 2013

Sheaf Cohomology

We need some basic homological algebra. Let $𝒞$ be an abelian category (for our purposes, $𝒞$ will be the category of $𝒪\text{-Modules,}$ where $𝒪$ is a sheaf of rings on a topological space $X\text{).}$ An object $I$ in $𝒞$ is injective if the functor $A\mapsto I\left(A\right)=\text{Hom}(A,I)$ is exact and not merely left exact: that is to say, whenever $A\to B$ is a monomorphism in $𝒞,$ the map $I\left(B\right)\to \left(IA\right)$ is surjective.

The category $𝒞$ has enough injectives if every object in $𝒞$ can be embedded in an injective object. Suppose that $𝒞$ has enough injectives, and let $A$ be an object in $𝒞\text{.}$ Then there exists an injective $I^0$ in $𝒞$ and a monomorphrsm $\mu :A\to I^0\text{.}$ Let $A^1=\text{Coker}\left(\mu \right),$ then there exists an injective $I^1$ in $𝒞$ and a monomorphism ${\mu }^1:A^1\to I^1\text{.}$ Let $A^2=\text{Coker}\left({\mu }^1\right),$ and so on. The short exact sequences $0\to A\to I^0\to A^1\to 0,$ $0\to A^1\to I^1\to A^2\to 0,$ etc., then stick together to form a long exact sequence:

$$\begin{array}{cc}\left(✶\right)& 0\to A\to I^0\stackrel{a^0}{⟶}{I}^1\stackrel{a^1}{⟶}{I}^2\stackrel{a^2}{⟶}\dots \end{array}$$

called an injective resolution of $A\text{.}$

Now let $F$ be a covariant additive left exact functor on $𝒞$ with values in an abelian category $𝒞\prime \text{.}$ If we operate on $\left(✶\right)$ with $F,$ we get a complex, so we can form its cohomology:

$$H^p=\text{Ker}\hspace{0.17em}F\left(a^p\right)/\text{Im}\hspace{0.17em}F\left(a^{p-1}\right)(p\ge 0\text{;}\hspace{0.17em}{a}^{-1}=0)\text{.}$$

The central fact is that $H^p$ depends (up to isomorphism) only on $F$ and $A$ and not on the injective resolution: it is denoted by $R^{p}F\left(A\right),$ and $R^{p}F$ is an additive functor, called-the $p\text{th}$ right derived functor of $F\text{.}$ Since $F$ is left exact, we have $R^{0}F=F\text{.}$

If $A$ is injective then $R^{p}F\left(a\right)=0$ for all $p>0\text{;}$ for $0\to A\to A\to 0$ is an injective resolution of $A\text{.}$

Theorem (8.1). If $0\to A\stackrel{\alpha }{\to }B\stackrel{\beta }{\to }C\to 0$ is an exact sequence in $𝒞,$ and if $F$ is a covariant additive left exact functor on $𝒞$ with values in an abelian category $𝒞\prime ,$ then there is an exact sequence in $𝒞\prime \text{:}$

$$0\to F\left(A\right)\stackrel{F\left(a\right)}{⟶}F\left(B\right)\stackrel{F\left(\beta \right)}{⟶}F\left(C\right)\stackrel{\partial }{⟶}{R}^{1}F\left(A\right)\stackrel{R^{1}F\left(a\right)}{⟶}{F}^{1}F\left(B\right)\stackrel{R^{1}F\left(\beta \right)}{⟶}{R}^{1}F\left(C\right)\stackrel{\partial }{⟶}{R}^{2}F\left(A\right)⟶\dots$$

For the definition of the 'coboundary morphisms' $\partial :R^{p-1}F\left(C\right)\to R^{p}F\left(A\right)$ and the proof of (8.1) we refer to Godement's book (or any book on homological algebra).

Grothendieck cohomology

We shall apply this machinery to the following situation: $(X,𝒪)$ is a ringed space and $𝒞$ is the category of $𝒪\text{-Modules.}$ Then $𝒞$ is abelian (as remarked in Chapter 7) and in fact $𝒞$ has enough injectives (proof e.g. in Godement's book). By (7.2), the section functor $\Gamma$ is a left exact functor on $𝒞$ with values in the category of $𝒪\left(X\right)\text{-modules.}$ The cohomology groups (which are in fact $𝒪\left(X\right)\text{-modules)}$ of $X$ with coefficients in the $𝒪\text{-Module}$ $ℱ$ are then defined to be

$$H^p(X,ℱ)=R^p\Gamma \left(ℱ\right)(p\ge 0)\text{.}$$

In particular, $H^0(X,ℱ)=\Gamma (X,ℱ)\text{.}$ From (8.1) we have an exact cohomology sequence: if $0\to ℱ\to 𝒢\to \mathscr{H}\to 0$ is an exact sequence of $𝒪\text{-Modules,}$ then the sequence

$$0⟶\Gamma (X,ℱ)⟶\Gamma (X,𝒢)⟶\Gamma (X,\mathscr{H})⟶H^1(X,ℱ)⟶H^1(X,𝒢)⟶H^1(X,\mathscr{H})⟶H^2(X,ℱ)⟶\dots$$

is exact.

This definition of the cohomology groups $H^p(X,ℱ)$ is due to Grothendieck.

$Č\text{ech}$ cohomology

There is an earlier definition of sheaf cohomology, modelled on $Č\text{ech}$ theory, which goes as follows. Let $𝓊={\left(U_i\right)}_{i\in I}$ be any open covering of $X\text{.}$ If $\sigma =(i_0,\dots ,i_p)$ is any $p\text{-simplex,}$ i.e, sequence of $p+1$ elements of the index set $I,$ let $U_{\sigma }$ denote the intersection $U_{i_0}\cap \dots \cap U_{i_p}\text{.}$ An (alternating) $p\text{-cochain}$ of the covering $𝓊$ with coefficients in the sheaf $ℱ$ is a function $c$ which associates with each $p\text{-simplex}$ $\sigma$ an element $c_{\sigma }\in ℱ\left(U_{\sigma }\right)$ in such a way that $c_{\sigma }$ is alternating in the indices $i_0,\dots ,i_p,$ and $c_{\sigma }=0$ whenever any two of the indices are equal. The $p\text{-cochains}$ form a group $C^p(𝓊,ℱ),$ which has a natural $𝒪\left(X\right)\text{-module}$ structure: if $a\in 𝒪\left(X\right),$ then ${\left(ac\right)}_{\sigma }$ is defined to be $\left(a\right|U_{\sigma })·c_{\sigma }\text{.}$ If we order the index set $I$ linearly then we may write $C^p(U,ℱ)=\prod _{\sigma }ℱ\left(U_{\sigma }\right),$ where in the product $\sigma$ runs over all $p\text{-simplexes}$ $(i_0,\dots ,i_p)$ such that $i_0<i_1<\dots <i_p\text{.}$

Define a coboundary homomorphism

$$d:C^p(𝓊,ℱ)⟶C^{p+1}(𝓊,ℱ)$$

as follows: if $c\in C^p(𝓊,ℱ),$ then

$${\left(dc\right)}_{i_0\dots i_{p+1}}=\sum _{k=0}^{p+1}{(-1)}^k{c}_{i_0\dots {\hat{i}}_k\dots i_{p+1}}|U_{i_0\dots i_{p+1}}$$

One verifies that $d^2=0\text{.}$ Thus $C^{•}(𝓊,ℱ)=\underset{p\ge 0}{⨁}{C}^p(𝓊,ℱ)$ is a complex of $𝒪\left(X\right)\text{-modules,}$ and we define the $p\text{th}$ cohomology group of the covering $𝓊$ with coefficients in $ℱ$ to be

$$H^p(𝓊,ℱ)=H^p\left(C^{•}(𝓊,ℱ)\right)(p>0)\text{.}$$

Next one shows that a refinement $𝓊\prime$ of $𝓊$ gives rise to well-defined homomorphisms $H^p(𝓊,ℱ)\to H^p(𝓊\prime ,ℱ)$ with the usual transitivity properties; these enable us to define the $Č\text{ech}$ cohomology groups of $X$ with coefficients in $ℱ\text{:}$

$${\check{H}}^p(X,ℱ)=\underset{\underset{𝓊}{⟶}}{\text{lim}}{H}^p(𝓊,ℱ),$$

the direct limit being taken over arbitrarily fine open coverings $𝓊$ of $X\text{.}$

The advantage of $Č\text{ech}$ cohomology is that one stands some chance of being able to compute it in given situations. The disadvantage, which is a serious one, is that the cohomology sequence in $Č\text{ech}$ cohomology is not necessarily exact. It is always the case that ${\check{H}}^0=H^0$ and ${\check{H}}^1=H^1$ (so that the $Č\text{ech}$ cohomology sequence is always exact as far as ${\check{H}}^1\text{)}$ but ${\check{H}}^p(X,ℱ)$ and $H^p(X,ℱ)$ are not necessarily the same for $p>1\text{.}$ There is a spectral sequence relating the two cohomologies (details in Godement's book), from which one can assert that ${\check{H}}^p=H^p$ for all $p$ under suitable hypotheses on $X$ or $ℱ$ or both. Here are two such 'comparison theorems':

Theorem (8.2). Let $𝓊$ be an open covering of $X,$ let $ℱ$ be a sheaf on $X,$ and suppose that, for all simplexes $\sigma =(i_0,\dots ,i_p),$ we have $H^p(U_{\sigma },ℱ|U_{\sigma })=0$ for all $q>0\text{.}$ Then $H^q(X,ℱ)=H^q(𝓊,ℱ)$ for all $q\ge 0\text{.}$

Theorem (8.3). (Cartan). Let $𝓊$ be an open covering of $X$ and $ℱ$ a sheaf on $X$ such that

(i)	$𝓊$ is closed under finite intersections;
(ii)	the sets of $𝓊$ form a basis of $X\text{;}$
(iii)	${\check{H}}^q(U,ℱ|U)=0$ for all $U\in 𝓊$ and all $q>0\text{.}$

Then ${\check{H}}^q(X,ℱ)\cong H^q(X,ℱ)$ for all $q\ge 0\text{.}$

Theorems (8.2) and (8.3) are proved in Godement's book. We shall sketch a proof of (8.2) avoiding the use of spectral sequences, but not (8.3). There are other comparison theorems: thus the conclusion of (8.3) is valid if $X$ is paracompact (and Hausdorff) and $ℱ$ is any sheaf of abelian groups. This one is of use if $X$ is a differentiable manifold or a complex manifold, but not in algebraic geometry.

The $Č\text{ech}$ resolution of a sheaf

Let $(X,𝒪)$ be a ringed space and $ℱ$ an $𝒪\text{-Module,}$ and let $𝓊={\left(U_i\right)}_{i\in I}$ be any open covering of $X\text{.}$ For each open set $V$ in $X$ let $V\cap 𝓊$ denote the open covering ${(V\cap U_i)}_{i\in I}$ of $V\text{.}$ Then we have $𝒪\left(V\right)\text{-modules}$ $C^p(V\cap 𝓊,ℱ|V)$ for each open set $V$ in $X$ and each integer $p\ge 0,$ hence presheaves $V\mapsto C^p(V\cap 𝓊,ℱ|V)$ for each $p\ge 0\text{.}$ These presheaves are easily verified to be sheaves; denote them by $C^p(𝓊,ℱ)\text{.}$ The coboundary operator $d:C^p\to C^{p+1}$ gives rise to sheaf homomorphisms

$$d:C^p(𝓊,ℱ)⟶C^{p+1}(𝓊,ℱ)\text{;}$$

also we have a sheaf homomorphism $j:ℱ\to C^0(𝓊,ℱ)$ defined as follows: if $s$ is a section of $ℱ$ over $V,$ then

$$j\left(s\right)={\left(s\right|V\cap U_i)}_{i\in I}\in C^0(V\cap 𝓊,ℱ|V)=C^0(𝓊,ℱ)\left(V\right)\text{.}$$

Proposition (8.4). The sequence

$$0⟶ℱ\stackrel{j}{⟶}{C}^0(𝓊,ℱ)\stackrel{d^0}{⟶}{C}^1(𝓊,ℱ)\stackrel{d^1}{⟶}\dots$$

is exact.

		Proof.
	(i) $j$ is a monomorphism. For if $j\left(s\right)=0$ then $s|V\cap V_i=0$ for all $i\in I,$ hence $s=0$ (since the $V\cap U_i$ cover V). (ii) $\text{Im}\left(j\right)=\text{Ker}\left(d^0\right)\text{.}$ Let $s=\left(s_i\right)\in C^0\left(V\right)\text{.}$ If $ds=0$ then ${\left(ds\right)}_{ij}=0$ for all pairs $(i,j)$ in $I,$ i.e. $s_i=s_j$ in $V\cap U_i\cap U_j\text{;}$ hence the $s_i$ fit together to give a section $\bar{s}$ of $ℱ$ over $V$ such that $\bar{s}|V\cap U_i=s_i$ for each $i\text{;}$ i.e. $s=j\left(\bar{s}\right)\text{.}$ Conversely, if $s=j\left(\bar{s}\right)$ for some $\bar{s}\in ℱ\left(V\right),$ then $s_i=\bar{s}|V\cap U_i,$ hence ${\left(ds\right)}_{ij}=s_j|V\cap U_i\cap U_j-s_i|V\cap U_i\cap U_j=0\text{.}$ (iii) $\text{Im}\left(d^{p-1}\right)=\text{Ker}\left(d^p\right)\text{.}$ We have $d^p\circ d^{p-1}=0,$ hence $\text{Im}\left(d^{p-1}\right)\subseteq \text{Ker}\left(d^p\right)\text{.}$ Conversely, let $u\in {\left(C^p(𝓊,ℱ)\right)}_x$ be such that $du=0\text{:}$ say $x\in U_i\text{.}$ Then there exists an open neighbourhood $V$ of $x$ contained in $U_i$ and an element $s\in C^p(𝓊,ℱ)\left(V\right)$ such that $s_x=u\text{.}$ If $\sigma =(i_0,\dots ,i_{p-1})$ is a $(p-1)\text{-simplex,}$ $i_{\sigma }$ let $i_{\sigma }$ denote the $p\text{-simplex}$ $(i,i_0,\dots ,i_{p-1})\text{.}$ We have $C^p(𝓊,ℱ)\left(V\right)=C^p(V\cap 𝓊,ℱ|V),$ hence $s$ is a family $\left(s_{\tau }\right)$ where $\tau$ runs through the $p\text{-simplexes}$ and $s_{\tau }\in ℱ(V\cap U_{\tau })\text{.}$ Define $t\in C^{p-1}(𝓊,ℱ)\left(V\right)$ by the rule $t_{\sigma }=s_{i\sigma }\in ℱ(V\cap U_i\cap U_{\sigma })=ℱ(V\cap U_{\sigma })\text{;}$ then $$\begin{array}{ccc}{\left(dt\right)}_{\tau }& =& \sum _{k=0}^p{(-1)}^k{t}_{{\tau }_k}|V\cap U_{\tau }({\tau }_k=k\text{th 'face' of}\hspace{0.17em}\tau )\\ & =& \sum _k{(-1)}^k{s}_{i{\tau }_k}|V\cap U_{\tau }\\ & =& s_{\tau }-{\left(ds\right)}_{i\tau }\\ & =& s_{\tau }\hspace{0.17em}\text{since}\hspace{0.17em}ds=0\text{.}\end{array}$$ Hence $dt=s$ and therefore $\text{Ker}\left(d^p\right)\subseteq \text{Im}\left(d^{p-1}\right)\text{.}$ $\square$

		Proof of (8.2).
	(i) Any product of injectives is injective (this is true in any abelian category). (ii) If $ℐ$ is an injective $𝒪\text{-Module}$ and $U$ is open in $X,$ then $ℐ|U$ is an injective $𝒪|U\text{-Module.}$ For we have ${\text{Hom}}_{𝒪|U}(𝒢,ℐ|U)={\text{Hom}}_{𝒪}({𝒢}^X,ℐ)$ for any $𝒪|U\text{-Module}$ $𝒢,$ where ${𝒢}^X$ denotes the sheaf on $X$ obtained by extending by zero outside $U\text{.}$ (iii) If $𝒥$ is an injective $𝒪|U\text{-Module}$ and $i:U\to X$ is the embedding of the open set $U$ in $X,$ then $i_{✶}𝒥$ is an injective $𝒪\text{-Module.}$ For we have ${\text{Hom}}_{𝒪}(ℱ,i_{✶}𝒥)\cong {\text{Hom}}_{𝒪|U}\left(ℱ\right|U,𝒥)$ for any $𝒪\text{-Module}$ $ℱ\text{.}$ (iv) With the notation of (8.4), we have $C^q(𝓊,ℱ)\left(V\right)=\prod _{\sigma }ℱ(V\cap U_{\sigma })=\prod _{\sigma }{i}_{{\sigma }^{*}}\left(ℱ\right|U_{\sigma })\left(V\right)$ where $\sigma$ runs through all $q\text{-simplexes}$ $(i_0,\dots ,i_q)$ such that $i_0<\dots <i_q$ (with respect to some linear ordering of the index set $I\text{)}$ and $i_{\sigma }$ is the embedding of $U_{\sigma }$ in $X\text{.}$ Hence if $ℱ$ is injective, then $C^q(𝓊,ℱ)$ is injective by (i), (ii) and (iii). (v) Let $0\to ℱ\to {ℐ}^0\to {ℐ}^1\to \dots$ be an injective resolution of $ℱ\text{.}$ Then for each simplex $\sigma$ the sequence $0\to ℱ|U_{\sigma }\to {ℐ}^0|U_{\sigma }\to {ℐ}^1|U_{\sigma }\to \dots$ is an injective resolution of $ℱ|U_{\sigma },$ by (iii) and the fact that restriction to an open set preserves exactness. Hence this sequence can be used to calculate the cohomology of $ℱ|U_{\sigma }\text{.}$ But by hypothesis $H^q(U_{\sigma },ℱ|U_{\sigma })=0$ for all $q>0\text{.}$ Hence the sequence $$0⟶ℱ\left(U_{\sigma }\right)⟶{ℐ}^0\left(U_{\sigma }\right)⟶{ℐ}^1\left(U_{\sigma }\right)⟶\dots$$ is exact. Hence, taking the product of these exact sequences for, all $q\text{-simplexes}$ $\sigma ,$ the sequence $$0⟶C^q(𝓊,ℱ)⟶C^q(𝓊,{ℐ}^0)⟶C^q(𝓊,{ℐ}^1)⟶\dots$$ is exact. (vi) Consider next the $Č\text{ech}$ resolution of ${ℐ}^p\text{:}$ $$0⟶{ℐ}^p⟶C^0(𝓊,{ℐ}^p)⟶C^1(𝓊,{ℐ}^p)⟶\dots$$ By (8.4) this is an exact sequence. By (iv) above, each $C^q(𝓊,{ℐ}^p)$ is injective, hence this is an injective resolution of ${ℐ}^p\text{;}$ but ${ℐ}^p$ has zero cohomology in dimensions $>0,$ hence the sequence $$0⟶{ℐ}^p\left(X\right)⟶C^0(𝓊,{ℐ}^p)\left(X\right)⟶C^1(𝓊,{ℐ}^p)\left(X\right)⟶\dots ,$$ that is to say the sequence $$0⟶{ℐ}^p\left(X\right)⟶C^0(𝓊,{ℐ}^p)⟶C^1(𝓊,{ℐ}^p)⟶\dots ,$$ is exact. (vii) We now have a double complex, in which all rows except for the top one, and all columns except for the left-hand one, are exact sequences (by (v) and (vi)): $$\begin{array}{ccccccc}& & 0& & 0& & 0\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& ⟶& ℱ\left(X\right)& ⟶& C^0(𝓊,ℱ)& ⟶& C^1(𝓊,ℱ)& ⟶& \dots \\ & & \downarrow & & \downarrow & & \downarrow \\ 0& ⟶& {ℐ}^0\left(X\right)& ⟶& C^0(𝓊,{ℐ}^0)& ⟶& C^1(𝓊,{ℐ}^0)& ⟶& \dots \\ & & \downarrow & & \downarrow & & \downarrow \\ 0& ⟶& {ℐ}^1\left(X\right)& ⟶& C^0(𝓊,{ℐ}^1)& ⟶& C^1(𝓊,{ℐ}^1)& ⟶& \dots \\ & & \downarrow & & \downarrow & & \downarrow \\ & & ⋮& & ⋮& & ⋮\end{array}$$ In such a situation the cohomology of the top row is isomorphic to the cohomology of the left-hand column. But in the present case the cohomology of the top row is the $Č\text{ech}$ cohomology $H^p(𝓊,ℱ),$ and the cohomology of the left-hand column is Grothendieck cohomology $H^p(X,ℱ)\text{.}$ $\square$

Notes and References

This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".

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