Contents

Idea

A proper base change theorem asserts a Beck-Chevalley condition for base change in cohomology along a proper map.

In étale cohomology

Let $p \colon X \longrightarrow S$ be a proper morphism of schemes. For $f \colon T \longrightarrow S$ any other morphism into $S$, consider the fiber product

$\array{ X \times_S T &\longrightarrow& X \\ {}^{\mathllap{f^\ast p}} \downarrow & & \downarrow^{\mathrlap{p}} \\ T &\stackrel{f}{\longrightarrow}& S \,. }$

The étale proper base change theorem says that in this situation and for $\mathcal{F}$ an abelian sheaf of torsion groups on $X$, the derived pull-push along the top and left is isomorphic to the derived push-pull along the bottom right. (…)