# Hypothesis under which sheaf cohomology is limit of Čech cohomology

From Mumford - Oda, Algebraic Geometry II, page 227

For general presheaves $\mathcal{F}$ [of abelian groups] and topological spaces $X$, one finally passes to the limit via $\mathrm{ref}$ over finer and finer coverings and defines: $$\mathrm{H}^i(X, \mathcal{F}) = \lim_{\stackrel{\longrightarrow}{\mathcal{U}}}\mathrm{H}^i(\mathcal{U}, \mathcal{F})$$

And there is a footnote

This group, the Čech cohomology, is often written $\check{\mathrm{H}}^i(X, \mathcal{F})$ to distinguish it from the “derived functor” cohomology. In most cases they are however canonically isomorphic and as we will not define the latter, we will not use the $\check{\phantom{\mathrm{H}}}$.

When $i=1$, this is proved true in Exercise III.4.4 of Hartshorne, Algebraic Geometry, but for higher $i$ there is no hint about when this is true.

It is also true for Hausdorff paracompact spaces, and there are several counter examples.

To specify a bit more, I am interested in the case when $$X=\mathrm{Spec}R$$ for some $R$ Noetherian commutative ring with identity, equipped with Zariski topology (so no Hausdorff here to help me).

So the main question is what hypothesis do I need on $R$, when $X=\mathrm{Spec}R$?

And, almost as a bonus, what are these “most cases”? Under which hypothesis do Čech cohomology and derived functor cohomology agree?
Since it is the most cases, these hypothesis should not be too restrictive, right?

There's indeed a very nice theorem showing the equivalence of Cech and derived functor cohomology.

The inductive limit Cech cohomology is seldom used in algebraic geometry exactly because is very badly behaved if the underlying space is not Hausdorff paracompact. It has very useful application in algebraic topology, instead.

In algebraic geometry one uses a slightly simpler version of Cech cohomology. Let $\mathfrak{U}=\{U_\alpha\}$ an open cover for a topological space $X$, such that $\alpha$ varies in an totally ordered set $(I,\leq)$. Let $\mathscr{F}$ be an abelian groups sheaf over $X$. Hence we define the Cech cochain complex as $$\check{C}^n(\mathfrak{U},\mathscr{F}):=\prod_{\alpha_0<\cdots<\alpha_n} \Gamma(U_{\alpha_0} \cap\ldots \cap U_{\alpha_n},\mathscr F )$$

It is the usual Cech complex, but computations are way simpler thanks to the total order. As a matter of fact, one can prove that this complex is homotopic to the usual Cech complex, so the Cech cohomology respect to $\mathfrak U$ is defined: $$\check{H}^n(\mathfrak{U},\mathscr F):= H^n(\check{C}^\bullet (\mathfrak{U},\mathscr{F}))$$

With this definition, you can find a lot of interesting properties that resemble those typical of sheaf cohomology. Then the result you're interested in.

Theorem. Let $X$ be a scheme and let be $\mathfrak U =\{U_i\}_{i\in I}$ an open cover for $X$ such that:

• $(I,\leq)$ is a totally ordered set;

• every $U_i$ is an affine open set in $X$;

• arbitrary intersections $U_{i_0}\cap \ldots \cap U_{i_n}$ are still affine open sets.

Therefore for every quasi-coherent sheaf $\mathscr F$ over $X$ the isomorphism $\check{H}^p(\mathfrak{U},\mathscr F)\simeq H^p(X,\mathscr{F})$ holds for every $p\geq 0$.

Details can be found in Goertz-Wedhorn Algebraic Geometry II book or, maybe more simply, in Kempf Algebraic varieties book.

• Hi! Well, since it is quasi-coherent and we have an affine covering, one can "simply" use Leary's theorem. But I am asking about sheaves of abelian groups, which are almost never quasi-coherent. Any hints on that? – dadexix86 Feb 29 '16 at 17:59
• This is not easy in these hypotheses, actually. The problem is that Zariski topology is too coarse to have a good behaviour in cohomology theories. The introduction of étale topology and sites was essential to have significant results. I don't know if you are familiar with such topics, I'm not really into the arguments. Hyper-coverings are another way to carry over Cech cohomology (you can find lots of references in Stacks project) – Caligula Feb 29 '16 at 18:23