January 10, 2011

The semicontinuity theorem

Posted by Akhil Mathew under algebraic geometry | Tags: , , |
[2] Comments 

 

Last time, we proved an important theorem. Namely, for a proper morphism of noetherian schemes X \to \mathrm{Spec} A, we showed that the cohomology of a coherent sheaf \mathcal{F} on X, flat over the base, could be described as the cohomology of a finite complex K of flat, finitely generated modules, and moreover that if we base-changed to some other scheme \mathrm{Spec B}, we just had to compute the cohomology of K \otimes_A B to get the cohomology of the base extension of the initial sheaf.

With this, it is not too hard to believe that as we vary over the fibers X_y, for y in the base, the cohomologies will have somewhat comparable dimensions. Or, at least, their dimensions will vary somewhat reasonably. The precise statement is provided by the semicontinuity theorem.

Theorem 6 (The Semicontinuity theorem) Let {f: X \rightarrow Y} be a proper morphism of noetherian schemes. Let {\mathcal{F}} be a coherent sheaf on {X}, flat over {Y}. Then the function {y \rightarrow \mathrm{dim} H^p(X_y, \mathcal{F}_y)} is upper semi-continuous on {Y}. Further, the function {y \rightarrow \sum_p (-1)^p H^p(X_y, \mathcal{F}_y)} is locally constant on {Y}. (more…)

 

January 9, 2011

The Grothendieck complex

Posted by Akhil Mathew under algebra, algebraic geometry, commutative algebra | Tags: , , |
1 Comment 

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 {f: X \rightarrow Y} be a proper morphism of noetherian schemes, {\mathcal{F}} a coherent sheaf on {X}. Suppose furthermore that {\mathcal{F}} is flat over {Y}; intuitively this means that the fibers {\mathcal{F}_y = \mathcal{F} \otimes_Y k(y)} form a “nice”  family of sheaves. In this case, we are interested in how the cohomology {H^p(X_y, \mathcal{F}_y) = H^p(X_y, \mathcal{F} \otimes k(y))} behaves as a function of {y}. 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 {Y = \mathrm{Spec} A} is affine, for some noetherian ring {A}. Consider an open affine cover {\left\{U_i\right\}} of {X}; we know, as {X} is separated, that the cohomology of {\mathcal{F}} on {X} can be computed using Cech cohomology. That is, there is a cochain complex {C^*(\mathcal{F})} of {A}-modules, associated functorially to the sheaf {\mathcal{F}}, such that

\displaystyle H^p(X, \mathcal{F}) = H^p(C^*(\mathcal{F})),

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

Proposition 1 Hypotheses as above, there exists a cochain complex {C^*(\mathcal{F})} of flat {A}-modules, associated functorially to {\mathcal{F}}, such that for any {A}-algebra {B} with associated morphism {f: \mathrm{Spec} B \rightarrow \mathrm{Spec} A}, we have\displaystyle H^p(X \times_A B, \mathcal{F} \otimes_A B) = H^p(C^*(\mathcal{F}) \otimes_A B).

Here, of course, we have abbreviated {X \times_A B} for the base-change {X \times_{\mathrm{Spec} A} \mathrm{Spec} B}, and {\mathcal{F} \otimes_A B} for the pull-back sheaf.

Proof: We have already given most of the argument. Now if {\left\{U_i\right\}} is an affine cover of {X}, then {\left\{U_i \times_A B \right\}} is an affine cover of the scheme {X \times_A B}. Furthermore, we have that

\displaystyle \Gamma(U_i \times_A B, \mathcal{F}\otimes_A B) = \Gamma(U_i , \mathcal{F}) \otimes_A B

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

\displaystyle \Gamma\left( \bigcap_{i \in I} U_i \times_A B, \mathcal{F}\otimes_A B\right) = \Gamma\left( \bigcap_{i \in I} U_i, \mathcal{F}\right) \otimes_A B. (more…)