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