User:TakuyaMurata/Proper base change theorem

There is also a proper base change theorem in topology. This article is not about it (yet).

In algebraic geometry, 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 $0\to K^{0}\to K^{1}\to \cdots K^{n}\to 0$ of finitely generated projective A-modules and a natural isomorphism of functors

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

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:

Corollary 1 (semicontinuity theorem): Let f and ${\mathcal {F}}$ as in the theorem. Then we have:

• (i) For each $p\geq 0$, the function $s\mapsto \dim _{k(s)}H^{p}({\mathcal {F}}_{s}):S\to \mathbb {Z}$ is locally constant.
• (ii) The function $s\mapsto \chi ({\mathcal {F}}_{s})$ is locally constant, where $\chi ({\mathcal {F}})$ denotes the Euler characteristic.

Corollary 2: Let f and ${\mathcal {F}}$ as in the theorem. Assume S is reduced and connected. Then for each $p\geq 0$ the following are equivalent

• (i) $s\mapsto \dim _{k(s)}H^{p}({\mathcal {F}}_{s})$ is constant.
• (ii) $R^{p}f_{*}{\mathcal {F}}$ is locally free and for all $s\in S$ 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 $s\in S$.

Corollary 3: Let f and ${\mathcal {F}}$ as in the theorem. 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 $s\in S$.

References

• Robin Hartshorne, Algebraic Geometry.
• David Mumford, Abelian Varieties.
• Vakil's notes