Yes, I’m still here. I just haven’t been in a blogging mood. I’ve been distracted a bit with the CRing project. I’ve also been writing a bunch of half-finished notes on Zariski’s Main Theorem and some of its applications, which I’ll eventually post.

I would now like to begin talking about the semicontinuity theorem in algebraic geometry, following Mumford’s *Abelian Varieties. *This result is used constantly throughout the book, mainly in showing that certain line bundles are trivial. Eventually, I’ll try to say something about this.

Let be a proper morphism of noetherian schemes, a coherent sheaf on . Suppose furthermore that is *flat* over ; intuitively this means that the fibers form a “nice” family of sheaves. In this case, we are interested in how the cohomology behaves as a function of . We shall see that it is upper semi-continuous and, under nice circumstances, its constancy can be used to conclude that the higher direct-images are locally free.

**1. The Grothendieck complex **

Let us keep the hypotheses as above, but assume in addition that is *affine*, for some noetherian ring . Consider an open affine cover of ; we know, as is separated, that the cohomology of on can be computed using Cech cohomology. That is, there is a cochain complex of -modules, associated functorially to the sheaf , such that

that is, sheaf cohomology is the cohomology of this cochain complex. Furthermore, since the Cech complex is defined by taking sections over the , we see that each term in is a *flat* -module as is flat. Thus, we have represented the cohomology of in a manageable form. We now want to generalize this to affine base-changes:

Proposition 1Hypotheses as above, there exists a cochain complex of flat -modules, associated functorially to , such that for any -algebra with associated morphism , we have

Here, of course, we have abbreviated for the base-change , and for the pull-back sheaf.

*Proof:* We have already given most of the argument. Now if is an affine cover of , then is an affine cover of the scheme . Furthermore, we have that

by definition of how the pull-backs are defined. Since taking intersections of the commutes with the base-change , we see more generally that for any finite set ,