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:

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

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

External links

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