Proper base change theorem

There is also a proper base change theorem in topology. For that, see base change map.

In algebraic geometry, there are at least two versions of proper base change theorems: one for ordinary cohomology and the other for étale cohomology.

In ordinary cohomology

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 $0 \to K^0 \to K^1 \to \cdots \to K^n \to 0$ 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 \ge 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: (the higher direct image $R^p f_* \mathcal{F}$ is coherent since f is proper.)

Corollary 1 (semicontinuity theorem): Let f and $\mathcal{F}$ as in the theorem (but S may not be affine). Then we have:

(i) For each $p \ge 0$ , the function $s \mapsto \dim_{k(s)} H^p (X_s, \mathcal{F}_s): S \to \mathbb{Z}$ is upper semicontinuous.
(ii) The function $s \mapsto \chi(\mathcal{F}_s)$ is locally constant, where $\chi(\mathcal{F})$ denotes the Euler characteristic.

Corollary 2: Assume S is reduced and connected. Then for each $p \ge 0$ the following are equivalent

(i) $s \mapsto \dim_{k(s)} H^p (X_s, \mathcal{F}_s)$ is constant.
(ii) $R^p f_* \mathcal{F}$ is locally free and 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

Furthermore, if these conditions hold, 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

Corollary 3: 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

In étale cohomology

In nutshell, the proper base change theorem states that the higher direct image $R^i f_* \mathcal{F}$ of a torsion sheaf $\mathcal{F}$ along a proper morphism f commutes with base change. A closely related, the finiteness theorem states that the étale cohomology groups of a constructible sheaf on a complete variety are finite. Two theorems are usually proved simultaneously.

Theorem (finiteness): Let X be a variety over a separably closed field and $\mathcal{F}$ a constructible sheaf on $X_\text{et}$ . Then $H^r(X, \mathcal{F})$ are finite in each of the following cases: (i) X is complete, or (ii) $\mathcal{F}$ has no p-torsion, where p is the characteristic of k.

References

Brian Conrad's handout
Robin Hartshorne, Algebraic Geometry. Theorem 12.11
Grothendieck, A. Éléments de géométrie algébrique. III. Etude cohomologique des faisceaux cohérents. II, Section 7.7
David Mumford, Abelian Varieties.
Vakil's notes
SGA 4
Milne, Étale cohomology
Gabber, "Finiteness theorems for étale cohomology of excellent schemes"

External links

http://mathoverflow.net/questions/90260/trouble-with-semicontinuity

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