Rayleigh–Bénard convection

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 $S=\mathrm{Spec}A$, 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

$H^p(X{\times }_S\mathrm{Spec}-,\mathcal{ℱ}{\otimes }_A-)\to H^p(K^{\bullet }{\otimes }_A-),p\ge 0$

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 $s\mapsto {\dim }_{k(s)}{H}^p(X_s,{\mathcal{ℱ}}_s):S\to \mathbb{\mathbb{Z}}$ 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

• Robin Hartshorne, Algebraic Geometry.
• David Mumford, Abelian Varieties.
• Vakil's notes
• SGA 4
• Milne, Étale cohomology
• Gabber, "“Finiteness theorems for étale cohomology of excellent schemes"