Proper base change theorem

In mathematics, the base change theorems relate the direct image and the pull-back of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves:

where

is a Cartesian square of topological spaces and ${\displaystyle {\mathcal {F}}}$ is a sheaf on X.

Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth.

A simple base change phenomenon arises in commutative algebra when A is a commutative ring and B and A' are two A-algebras. Let ${\displaystyle B'=B\otimes _{A}A'}$. In this situation, given a B-module M, there is an isomorphism (of A' -modules):

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