The next big application of the Koszul complex and this general machinery that I have in mind is to projective space. Namely, consider a ring {A}, and an integer {n \in \mathbb{Z}_{\geq 0}}. We have the {A}-scheme {\mathop{\mathbb  P}^n_A = \mathrm{Proj} A[x_0, \dots, x_n]}. Recall that on it, we have canonical line bundles {\mathcal{O}(m)} for each {m \in \mathbb{Z}}, which come from homogeneous localization of the {A[x_0, \dots,  x_n]}-modules obtained from {A[x_0, \dots, x_n]} itself by twisting the degrees by {m}. When {A} is a field, the only line bundles on it are of this form. (I am not sure if this is true in general. I think it will be true, but perhaps someone can confirm.)

It will be useful to compute the cohomology of these line bundles. For one thing, this will lead to Serre duality, from a very convenient isomorphism that will spring up. For another, we will see that they are finitely generated over {A}. This is far from obvious. The scheme {\mathop{\mathbb P}^n_A} is not finite over {A}, and a priori this is not expected.

But to start, let’s think more abstractly. Let {X} be any quasi-compact, quasi-separated scheme; we’ll assume this for reasons below. Let {\mathcal{L}} be a line bundle on {X}, and {\mathcal{F}} an arbitrary quasi-coherent sheaf. We can consider the twists {\mathcal{F} \otimes \mathcal{L}^{\otimes m}} for any {m \in  \mathbb{Z}}. This is a bunch of sheaves, but it is something more.

Let us package these sheaves together. Namely, let us consider the sheaves:

\displaystyle  \bigoplus  \mathcal{L}^{\otimes m}, \quad \mathcal{H}=\bigoplus \mathcal{F} \otimes  \mathcal{L}^{\otimes m}

Here the former is a quasi-coherent sheaf of algebras, while the latter is a quasi-coherent sheaf of modules over the sheaf of algebras. The point is that this extra structure will come in handy while computing cohomology. Then to compute the cohomology of {\mathcal{F}}, we just read off the right piece. Let {S = \Gamma(X, \bigoplus \mathcal{L}^{\otimes  m})}; it is a ring, and indeed a graded ring, because by quasi-compactness {S =\bigoplus  \Gamma(X, \mathcal{L}^{\otimes  m})}. Since {S} acts on the sheaf {\mathcal{H}}—in fact, {\mathcal{H}} is a sheaf of graded {\mathcal{H}}-modules, we see that

\displaystyle  H^k(X,\mathcal{H})

is a graded {S}-module for any {k}. Now suppose further that {f_1, \dots, f_r \in \Gamma(X,  \mathcal{L})}.

We are going to use these to define a collection {X_{f_i} \subset  X}, as we did when talking about the Koszul complex and Cech cohomology. Now {X_{f_i}} will denote the set of points where {f_i} does not generate the stalk of {\mathcal{L}}; this is clearly the generalization when the {f_i} are functions.

The point is that we are going to play the same game with these {f_i} and show that Cech cohomology is Koszul cohomology. We need a few preliminaries. Fix {f \in \mathcal{L}} arbitary. Since {X} is quasi-compact and quasi-separated, we have the following (it’s in Hartshorne II.5 or somewhere in EGA I.9, at least for the old EGA I). Namely, if {s \in  \mathcal{F}(X_f)}, there is a tensor power {s  \otimes f^N} which extends to {X}. Also, if {s \in  \mathcal{F}(X)} is zero on {X_f}, then a power {s \otimes  f^N} is zero in {\mathcal{F}(X)}. What this all says is that

Proposition 43 As a graded {S}-module, we have\displaystyle  \Gamma(X_f, \mathcal{H}) = \Gamma(X, \mathcal{H} )_f.

This is awfully similar to what we had earlier when {\mathcal{L} = \mathcal{O}_X} and {f} was a global regular function. Here, however, it is really necessary to consider all the “twists” {\mathcal{F} \otimes \mathcal{L}^{\otimes m}} to even make sense of the localization. So the business is slightly more complicated. But is unavoidable when we want to deal with projective space. There are very few global sections of the structure sheaf, but there are sections of certain invertible sheaves.

Let us now compute the Cech complex {C_*} for {\mathcal{H}} with respect to the open cover {\left\{X_{f_i}\right\}}. Let {M = \Gamma(X, \mathcal{H})}—the previous result suggests that this will play an important role in the sequel. This is an {S}-module.

Proposition 44 For each {k}, the {k}th term {C_k} of the Cech complex {C_*} is isomorphic to the {k+1}st term of the direct limit of the Koszul complexes {\varinjlim_t K_*(f_1^t, \dots,  f_r^t, M)} as a graded object.

This is a big deal because it shows that we can apply the same reasoning as before! On the other hand, it also means that the argument is really the same—the fact that localization corresponds to restricting to {X_{f_i}}, the interpretation of localization as a direct limit, and the checking that the Cech boundary is the Koszul boundary. The fact that this is a graded isomorpism comes from paying attention to how the gradings work out–note that the multiplication maps in the direct limit do change the gradings. I am reluctant to spend too much time on picking technical nits since this is a blog; if you’re curious, it’s in EGA III.2.

I will thus omit the proof.

Advertisements