## Chapter 8

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 $A↦I\left(A\right)=\text{Hom}\left(A,I\right)$ 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:

$(✶) 0→A→I0 ⟶a0I1 ⟶a1I2 ⟶a2…$

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:

$Hp=Ker F(ap) /Im F(ap-1) (p≥0; a-1=0) .$

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 $𝒞\prime \text{:}$

$0→F(A) ⟶F(a)F(B) ⟶F(β)F(C) ⟶∂R1F(A) ⟶R1F(a) F1F(B) ⟶R1F(β) R1F(C) ⟶∂R2F(A) ⟶…$

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: $\left(X,𝒪\right)$ 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

$Hp(X,ℱ)=Rp Γ(ℱ) (p≥0).$

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

$0⟶Γ(X,ℱ)⟶ Γ(X,𝒢)⟶ Γ(X,ℋ)⟶ H1(X,ℱ)⟶ H1(X,𝒢)⟶ H1(X,ℋ)⟶ H2(X,ℱ)⟶…$

is exact.

This definition of the cohomology groups ${H}^{p}\left(X,ℱ\right)$ 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 $\sigma =\left({i}_{0},\dots ,{i}_{p}\right)$ 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 ${C}^{p}\left(𝓊,ℱ\right),$ 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 $\left(a|{U}_{\sigma }\right)·{c}_{\sigma }\text{.}$ If we order the index set $I$ linearly then we may write ${C}^{p}\left(U,ℱ\right)=\prod _{\sigma }ℱ\left({U}_{\sigma }\right),$ where in the product $\sigma$ runs over all $p\text{-simplexes}$ $\left({i}_{0},\dots ,{i}_{p}\right)$ such that ${i}_{0}<{i}_{1}<\dots <{i}_{p}\text{.}$

Define a coboundary homomorphism

$d:Cp(𝓊,ℱ)⟶ Cp+1(𝓊,ℱ)$

as follows: if $c\in {C}^{p}\left(𝓊,ℱ\right),$ then

$(dc) i0…ip+1 =∑k=0p+1 (-1)k c i0…i^k …ip+1 | Ui0…ip+1$

One verifies that ${d}^{2}=0\text{.}$ Thus ${C}^{•}\left(𝓊,ℱ\right)=\underset{p\ge 0}{⨁}{C}^{p}\left(𝓊,ℱ\right)$ 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

$Hp(𝓊,ℱ)=Hp (C•(𝓊,ℱ)) (p>0).$

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

$Hˇp(X,ℱ)= lim⟶𝓊Hp (𝓊,ℱ),$

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 ${\stackrel{ˇ}{H}}^{0}={H}^{0}$ and ${\stackrel{ˇ}{H}}^{1}={H}^{1}$ (so that the $Č\text{ech}$ cohomology sequence is always exact as far as ${\stackrel{ˇ}{H}}^{1}\text{)}$ but ${\stackrel{ˇ}{H}}^{p}\left(X,ℱ\right)$ and ${H}^{p}\left(X,ℱ\right)$ 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 ${\stackrel{ˇ}{H}}^{p}={H}^{p}$ 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 $\sigma =\left({i}_{0},\dots ,{i}_{p}\right),$ we have ${H}^{p}\left({U}_{\sigma },ℱ|{U}_{\sigma }\right)=0$ for all $q>0\text{.}$ Then ${H}^{q}\left(X,ℱ\right)={H}^{q}\left(𝓊,ℱ\right)$ 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) ${\stackrel{ˇ}{H}}^{q}\left(U,ℱ|U\right)=0$ for all $U\in 𝓊$ and all $q>0\text{.}$

Then ${\stackrel{ˇ}{H}}^{q}\left(X,ℱ\right)\cong {H}^{q}\left(X,ℱ\right)$ 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 $\left(X,𝒪\right)$ 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 ${\left(V\cap {U}_{i}\right)}_{i\in I}$ of $V\text{.}$ Then we have $𝒪\left(V\right)\text{-modules}$ ${C}^{p}\left(V\cap 𝓊,ℱ|V\right)$ for each open set $V$ in $X$ and each integer $p\ge 0,$ hence presheaves $V↦{C}^{p}\left(V\cap 𝓊,ℱ|V\right)$ for each $p\ge 0\text{.}$ These presheaves are easily verified to be sheaves; denote them by ${C}^{p}\left(𝓊,ℱ\right)\text{.}$ The coboundary operator $d:{C}^{p}\to {C}^{p+1}$ gives rise to sheaf homomorphisms

$d:Cp(𝓊,ℱ)⟶ Cp+1(𝓊,ℱ);$

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

$j(s)= (s|V∩Ui)i∈I ∈C0 (V∩𝓊,ℱ|V)= C0(𝓊,ℱ)(V).$

Proposition (8.4). The sequence

$0⟶ℱ⟶jC0 (𝓊,ℱ)⟶d0 C1(𝓊,ℱ) ⟶d1…$

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 $\left(i,j\right)$ 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 $\stackrel{‾}{s}$ of $ℱ$ over $V$ such that $\stackrel{‾}{s}|V\cap {U}_{i}={s}_{i}$ for each $i\text{;}$ i.e. $s=j\left(\stackrel{‾}{s}\right)\text{.}$ Conversely, if $s=j\left(\stackrel{‾}{s}\right)$ for some $\stackrel{‾}{s}\in ℱ\left(V\right),$ then ${s}_{i}=\stackrel{‾}{s}|V\cap {U}_{i},$ 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 $u\in {\left({C}^{p}\left(𝓊,ℱ\right)\right)}_{x}$ 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 $s\in {C}^{p}\left(𝓊,ℱ\right)\left(V\right)$ such that ${s}_{x}=u\text{.}$ If $\sigma =\left({i}_{0},\dots ,{i}_{p-1}\right)$ is a $\left(p-1\right)\text{-simplex,}$ ${i}_{\sigma }$ let ${i}_{\sigma }$ denote the $p\text{-simplex}$ $\left(i,{i}_{0},\dots ,{i}_{p-1}\right)\text{.}$ We have ${C}^{p}\left(𝓊,ℱ\right)\left(V\right)={C}^{p}\left(V\cap 𝓊,ℱ|V\right),$ hence $s$ is a family $\left({s}_{\tau }\right)$ where $\tau$ runs through the $p\text{-simplexes}$ and ${s}_{\tau }\in ℱ\left(V\cap {U}_{\tau }\right)\text{.}$ Define $t\in {C}^{p-1}\left(𝓊,ℱ\right)\left(V\right)$ by the rule ${t}_{\sigma }={s}_{i\sigma }\in ℱ\left(V\cap {U}_{i}\cap {U}_{\sigma }\right)=ℱ\left(V\cap {U}_{\sigma }\right)\text{;}$ then $(dt)τ = ∑k=0p (-1)ktτk| V∩Uτ (τk=kth 'face' of τ) = ∑k(-1)k siτk| V∩Uτ = sτ-(ds)iτ = sτ since ds=0.$ 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 ${\text{Hom}}_{𝒪|U}\left(𝒢,ℐ|U\right)={\text{Hom}}_{𝒪}\left({𝒢}^{X},ℐ\right)$ 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 ${\text{Hom}}_{𝒪}\left(ℱ,{i}_{✶}𝒥\right)\cong {\text{Hom}}_{𝒪|U}\left(ℱ|U,𝒥\right)$ for any $𝒪\text{-Module}$ $ℱ\text{.}$ (iv) With the notation of (8.4), we have ${C}^{q}\left(𝓊,ℱ\right)\left(V\right)=\prod _{\sigma }ℱ\left(V\cap {U}_{\sigma }\right)=\prod _{\sigma }{i}_{{\sigma }^{*}}\left(ℱ|{U}_{\sigma }\right)\left(V\right)$ where $\sigma$ runs through all $q\text{-simplexes}$ $\left({i}_{0},\dots ,{i}_{q}\right)$ 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 ${C}^{q}\left(𝓊,ℱ\right)$ 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 ${H}^{q}\left({U}_{\sigma },ℱ|{U}_{\sigma }\right)=0$ for all $q>0\text{.}$ Hence the sequence $0⟶ℱ(Uσ)⟶ ℐ0(Uσ)⟶ ℐ1(Uσ)⟶…$ is exact. Hence, taking the product of these exact sequences for, all $q\text{-simplexes}$ $\sigma ,$ the sequence $0⟶Cq(𝓊,ℱ)⟶ Cq(𝓊,ℐ0) ⟶Cq(𝓊,ℐ1) ⟶…$ is exact. (vi) Consider next the $Č\text{ech}$ resolution of ${ℐ}^{p}\text{:}$ $0⟶ℐp⟶C0 (𝓊,ℐp)⟶ C1(𝓊,ℐp) ⟶…$ By (8.4) this is an exact sequence. By (iv) above, each ${C}^{q}\left(𝓊,{ℐ}^{p}\right)$ is injective, hence this is an injective resolution of ${ℐ}^{p}\text{;}$ but ${ℐ}^{p}$ has zero cohomology in dimensions $>0,$ hence the sequence $0⟶ℐp(X)⟶ C0(𝓊,ℐp)(X) ⟶C1(𝓊,ℐp) (X)⟶…,$ that is to say the sequence $0⟶ℐp(X)⟶ C0(𝓊,ℐp) ⟶C1(𝓊,ℐp) ⟶…,$ 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)): $0 0 0 ↓ ↓ ↓ 0 ⟶ ℱ(X) ⟶ C0(𝓊,ℱ) ⟶ C1(𝓊,ℱ) ⟶ … ↓ ↓ ↓ 0 ⟶ ℐ0(X) ⟶ C0(𝓊,ℐ0) ⟶ C1(𝓊,ℐ0) ⟶ … ↓ ↓ ↓ 0 ⟶ ℐ1(X) ⟶ C0(𝓊,ℐ1) ⟶ C1(𝓊,ℐ1) ⟶ … ↓ ↓ ↓ ⋮ ⋮ ⋮$ 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 ${H}^{p}\left(𝓊,ℱ\right),$ and the cohomology of the left-hand column is Grothendieck cohomology ${H}^{p}\left(X,ℱ\right)\text{.}$ $\square$

## Notes and References

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