User:TakuyaMurata/Proper base change theorem


 * There is also a proper base change theorem in topology. This article is not about it (yet).

In algebraic geometry, the proper base change theorem states the following: let $$f: X \to S$$ be a proper morphism between noetherian schemes, and $$\mathcal{F}$$ S-flat coherent sheaf on $$X$$. If $$S = \operatorname{Spec} A$$, then there is a finite complex $$0 \to K^0 \to K^1 \to \cdots K^n \to 0$$ of finitely generated projective A-modules and a natural isomorphism of functors
 * $$H^p(X \times_S \operatorname{Spec} -, \mathcal{F} \otimes_A -) \to H^p(K^\bullet \otimes_A -), p > 0$$

on the category of $$A$$-algebras.

There are several corollaries to the theorem, some of which are also referred to as proper base change theorems:

Corollary 1 (semicontinuity theorem): Let f and $$\mathcal{F}$$ as in the theorem. Then we have:
 * (i) For each $$p \ge 0$$, the function $$s \mapsto \dim_{k(s)} H^p (\mathcal{F}_s): S \to \mathbb{Z}$$ is locally constant.
 * (ii) The function $$s \mapsto \chi(\mathcal{F}_s)$$ is locally constant, where $$\chi(\mathcal{F})$$ denotes the Euler characteristic.

Corollary 2: Let f and $$\mathcal{F}$$ as in the theorem. Assume S is reduced and connected. Then for each $$p \ge 0$$ the following are equivalent
 * (i) $$s \mapsto \dim_{k(s)} H^p (\mathcal{F}_s)$$ is constant.
 * (ii) $$R^p f_* \mathcal{F}$$ is locally free and for all $$s \in S$$ the natural map
 * $$R^p f_* \mathcal{F} \otimes_{\mathcal{O}_S} k(s) \to H^p(X_s, \mathcal{F}_s)$$
 * is an isomorphism for all $$s \in S$$.

Corollary 3: Let f and $$\mathcal{F}$$ as in the theorem. Assume that for some p $$H^p(X_s, \mathcal{F}_s) = 0$$ for all $$s \in S$$. Then the natural map
 * $$R^{p-1} f_* \mathcal{F} \otimes_{\mathcal{O}_S} k(s) \to H^{p-1}(X_s, \mathcal{F}_s)$$
 * is an isomorphism for all $$s \in S$$.