Cohomology in Algebraic Geometry

In Homology and Cohomology we discussed cohomology as used to study topological spaces. In this post we study cohomology in the context of algebraic geometry. We will need the concepts we have discussed in Presheaves, Sheaves, More on Chain Complexes, and The Hom and Tensor Functors.

Sheaf cohomology is simply the derived functor (see More on Chain Complexes and The Hom and Tensor Functors) of the global section functor $\Gamma (X,-)$ , which assigns to a sheaf $\mathcal{F}$ its set of global sections $\Gamma (X,\mathcal{F})=\mathcal{F}(X)$ .

One thing to note here, in constructing our resolutions, is that we are dealing with sheaves of modules, instead of just modules. The morphisms of sheaves of modules on a topological space $X$ are defined as homomorphisms of modules $\mathcal{F}(U)\rightarrow \mathcal{G}(U)$ for every open set $U$ of $X$ .

Since our definition is quite abstract, we also discuss here Cech cohomology, which is more concrete. Let $X$ be a topological space, and let $\mathfrak{U}=(U_{i})_{i\in I}$ be an open covering of $X$ . We let

$\displaystyle C^{p}(\mathfrak{U},\mathcal{F})=\prod_{i_{0}<...<i_{p}}\mathcal{F}(U_{i_{0},...,i_{p}})$

where

$\displaystyle \mathcal{F}(U_{i_{0},...,i_{p}})=U_{i_{0}}\cap ...\cap U_{i_{p}}$

The coboundary maps $d^{p}:C^{p}\rightarrow C^{p+1}$ are given by

$\displaystyle (d^{p}\alpha)_{i_{0},...,i_{p+1}}=\sum_{k=0}^{p+1}(-1)^{k}\alpha_{i_{0},...,\hat{i_{k}},...,i_{p+1}}|_{U_{i_{0},...,i_{p}}}$

where $\hat{i_{k}}$ means that the index $i_{k}$ is to be omitted. The Cech cohomology is then given by the cohomology of this complex.

The Cech cohomology is equivalent to the sheaf cohomology, if the sheaf is quasi-coherent (see More on Sheaves). The injective resolution of the sheaf $\mathcal{F}$ is given by a “sheafified” version of the chain complex we constructed earlier. For an open subset $V$ of $X$ , let $f:V\rightarrow X$ be the inclusion map. We define

$\displaystyle \mathcal{C}^{p}(\mathfrak{U},\mathcal{F})=\prod_{i_{0}<...<i_{p}}f_{*}(\mathcal{F}|_{U_{i_{0},...,i_{p}}})$

with the same definition for the coboundary map as earlier. Then

$\Gamma(X,\mathcal{C}^{p}(\mathfrak{U},\mathcal{F}))=C^{p}(\mathfrak{U},\mathcal{F})$

from which it can be seen that the Cech cohomology is indeed the derived functor of the global section functor $\Gamma (X,-)$ .

Up to now, for our topological spaces we have always used the Zariski topology. We now introduce another kind of topology called the Etale topology. The Etale topology is not a topology in the sense of Basics of Topology and Continuous Functions, but a Grothendieck topology, which we have discussed in More Category Theory: The Grothendieck Topos. Our underlying category will be written $\text{Et}/X$ , and its objects will be etale morphisms (to be explained later) $h:U\rightarrow X$ , and its morphisms will be etale morphisms $f:U\rightarrow U'$ such that if $g:U'\rightarrow X$ then $f\circ g=h$ .

An etale morphism is a morphism of schemes that is both flat (see The Hom and Tensor Functors) and unramified. A morphism $f:Y\rightarrow X$ is said to be unramified if for all points $y$ of $Y$ the morphism $\mathcal{O}_{X,f(y)}\rightarrow\mathcal{O}_{Y,y}$ of local rings (see Localization) has the property that $\mathfrak{m}_{Y,y}=\mathfrak{m}_{X,f(y)}\cdot \mathcal{O}_{Y,y}$ and the residue field $\mathcal{O}_{Y,y}/\mathfrak{m}_{Y,y}$ is a finite separable field extension of $\mathcal{O}_{X,f(y)}/\mathfrak{m}_{X,f(y)}$ .

The concept of field extensions was discussed in Algebraic Numbers. We have explored in that same post how field extensions $F\subset K$ may be “generated” by the roots of polynomials with coefficients in $F$ . The field extension is called separable if the aforementioned polynomial (called the minimal polynomial) has distinct roots.

An unramified morphism $f:Y\rightarrow X$ is also required to be locally of finite type, which means that for every open subset $\text{Spec }A$ (we recall that in Localization we have updated our definition of schemes to mean something that “locally” looks like our “old” definition of schemes – the mathematical objects referred to by this “old” definition will henceforth be referred to as affine schemes) of $X$ and every open subset $\text{Spec }B$ of $f^{-1}(\text{Spec }A)$ the induced morphism $A\rightarrow B$ makes $B$ into a “finitely generated” $A$ -algebra.

Using the etale topology to define the sheaves to be used in cohomology results in etale cohomology, the original driving force for the development of the concept of the Grothendieck topos. Hopefully we will be able to flesh out more of this interesting theory in future posts.

References:

Sheaf Cohomology on Wikipedia

Cech Cohomology on Wikipedia

Etale Morphism on Wikipedia

Etale Cohomology on Wikipedia

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Lectures on Etale Cohomology by J.S. Milne

Algebraic Geometry by Robin Hartshorne