nLab proper base change theorem



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

In étale cohomology

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

X× ST X f *p p T f S. \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 XX, the derived pull-push along the top and left is isomorphic to the derived push-pull along the bottom right. (…)

(Milne, theorem 17.10)


Last revised on November 24, 2013 at 06:51:44. See the history of this page for a list of all contributions to it.