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 , and an integer . We have the -scheme . Recall that on it, we have canonical line bundles for each , which come from homogeneous localization of the -modules obtained from itself by twisting the degrees by . When 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 . This is far from obvious. The scheme is not finite over , and a priori this is not expected.
But to start, let’s think more abstractly. Let be any quasi-compact, quasi-separated scheme; we’ll assume this for reasons below. Let be a line bundle on , and an arbitrary quasi-coherent sheaf. We can consider the twists for any . This is a bunch of sheaves, but it is something more.
Let us package these sheaves together. Namely, let us consider the sheaves: