It’s funny that the topics I planned to discuss on this blog in the next month have turned out to largely match the topics covered in my commutative algebra class. As a result, I think I will move the focus closer to algebraic geometry. Today, I shall explain how the connection explained yesterday between Koszul and Cech cohomology lets us compute the cohomology of an affine space. This is the proof that Grothendieck gives in EGA III, and it has the advantage (unlike the argument in Hartshorne, though I’ll probably later end up explaining that too) of applying to non-noetherian rings. It is a nice application of the basic properties of the Koszul complex, though it has the disadvantage of requiring a rather non-elementary result in sheaf theory (which I’ll discuss next).

0.10. The cohomology of affine space

We are now going to prove the first fundamental theorem on the cohomology of quasi-coherent sheaves:

Theorem 35 (Cohomology of an affine) Let {R} be a ring, and let {\mathcal{F}} be a quasi-coherent sheaf on {X=\mathrm{Spec} R}. Then\displaystyle H^k( X, \mathcal{F})=0, \quad k \geq 1.

I have earlier discussed a proof due to Kempf. What we will now sketch is a much less elementary and significantly more complicated argument. Nonetheless, it has the virtue of being general, and telling us something about projective space too, as we shall see eventually.

Proof: This proof proceeds first by analyzing the Cech cohomology. We will show that this is zero. Then, we shall appeal to some general sheaf-theoretic business to prove the result for standard cohomology. In particular, we are going to prove:

Theorem 36 Let {\mathcal{F}} be a quasi-coherent sheaf on {X=\mathrm{Spec} R}. Let {\left\{f_i\right\} \subset R} be a finite set of elements generating the unit ideal. Then the higher Cech cohomology of {\mathcal{F}} with respect to the open cover {D(f_i)} vanishes.

Proof: Let {M = \Gamma(X, \mathcal{F})}; then {\mathcal{F}} is the sheaf {\widetilde{M}} on {\mathrm{Spec} R}, by Hartshorne’s chapter II terminology. Now {\mathrm{Spec} R} is a quasi-compact, quasi-separated scheme. (It’s even separated!) Moreover, the {D(f_i)} are equal to what we called the {X_{f_i}}—the sets of “nonvanishing” of the global “functions” {f_i \in \Gamma(X, \mathcal{O}_X)}.

In particular, the earlier result about Cech cohomology goes in force. The {k}th Cech cohomology (for {k \geq 1}) of {\mathcal{F}} with respect to {\left\{D(f_i)\right\}} is the direct limit of the Koszul cohomologies,

\displaystyle H^k(\mathfrak{A}, \mathcal{F})=\varinjlim_m K^{k+1}(\mathbf{f}^m , M).

But {\mathbf{f}} generates the unit ideal, and so does {\mathbf{f}^m} as a result—this is a general fact about Koszul cohomology, which I should talk about soon. Thus the Koszul cohomology is trivial, and we find

\displaystyle H^k(\mathfrak{A}, \mathcal{F}) = 0

for {k \geq 1}. This proves the result. The rest of this proof requires some work. Namely, we are going to have to show the following useful result of Cartan:

Theorem 37 (Cartan) Let {X} be a space, {\mathcal{F}} a sheaf on {X}. Suppose there is a family of open sets {\mathfrak{A} } of {X} such that the following conditions are satisfied. First, {\mathfrak{A}} is a basis, and it is closed under finite intersections. If {\mathfrak{B} \subset \mathfrak{A}} is a finite open covering of {U \in \mathfrak{A}}, then the Cech cohomology in positive dimension vanishes,\displaystyle H^k(\mathfrak{B}, \mathcal{F})=0.

Then the usual cohomology vanishes equals the Cech cohomology: \displaystyle H^k(X, \mathcal{F}) =H^k(\mathfrak{A}, \mathcal{F}), \ k \geq 1.

The theorem of Cartan immediately implies the vanishing of quasi-coherent cohomology on an affine from what we have seen. Indeed, we use the basic open sets {D(f)} as the basis for {\mathrm{Spec} A}; this is closed under intersection. We have seen that the Cech cohomology of this (on any open subset) vanishes. Hence the same is true for derived functor cohomology. The proof of Cartan’s theorem is a long story in itself, involving a spectral sequence between Cech and usual cohomology, and I shall defer it.