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 $S=\operatorname {Spec} A$, 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

$H^{p}(X\times _{S}\operatorname {Spec} -,{\mathcal {F}}\otimes _{A}-)\to H^{p}(K^{\bullet }\otimes _{A}-),p\geq 0$

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 $p\geq 0$, 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 $p\geq 0$ 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"