I want to share a proof of the Cohomology and Base Change theorem that really makes it much much more clearer to me than it had been before. There might be some uncertainty as to its origin, so I’ll give my take on it. I heard it from Eric Larson a couple of years ago. We had discussed the fact that it should be some easy-to-understand fact about maps of free modules over Noetherian rings. (Many people have known this — Max Lieblich and I had discussed it before; it is alluded to in Nitsure’s excellent presentation in FGA Explained; and I am certain similar discussions have happened many times in the past.)

Eric went home and figured it out, and the next day sent me a short and sweet argument in a pdf file (see below). He had thought he had heard it before, but the only plausible sources he could have heard it from (Joe Harris, or the argument in Eisenbud and Harris) don’t have the argument. But he had certainly heard an argument for Grauert’s Theorem (where the base is reduced). So until I learn otherwise, I suspect he just thought he had heard a full argument of Cohomology and Base Change, and then tried to reconstruct how it should go, and just figured it out. In other words, for now, I believe the argument is original to him. But I would have expected that such a simple argument (especially with its central insight) would have been independently discovered earlier, perhaps many times. However, I am not aware of it in the literature anywhere. Can anyone tell me where it has appeared before? It is such a simple argument that, if I had not been consciously looking for it earlier, I would have believed that I had surely known it myself. So if someone says “I knew that, I just never wrote it down”, I’m not going to value that too highly. On the other hand, there are a number of great explanations of things in algebraic geometry that Dennis Gaitsgory told undergraduates when he was at Harvard; and Nitsure’s argument is presumably written down somewhere; so I know there are things I haven’t seen.

(Incidentally, Ogus told me a fantastically elegant proof of the local criterion for flatness, which was in his first published paper I think, that I love.)

Of course, I’ve digested Eric’s argument into The Rising Sea (and I really wish I hadn’t moved around part of a chapter a couple of years ago, leaving the manuscript in some disarray; I hope to post the new version before too too long). And I’ll eventually post my take on Ogus’ argument here, and I’m also digesting it into the Rising Sea.

One of Eric’s insights, for me, is this. When we generalize the notion of “finite-dimensional vector space” to families, we get finite-rank vector bundles. These have constant rank, but for a coherent sheaf to be a (finite rank) vector bundle, we need more than it be constant rank. Eric suggests that to generalize the notion of “map of finite-dimemnsional vector spaces” in a particularly good way, we want more than a map of vector bundles (as coherent sheaves) — and we want more than that it is of “constant rank”. It is this stronger condition about maps of vector bundles that gives this strong condition of cohomology commuting with base change, which can be applied in different settings (including Grauert, and the Cohomology and Base Change Theorem).