Base change theorems

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

${\displaystyle g^{*}(R^{r}f_{*}{\mathcal {F}})\to R^{r}f'_{*}(g'^{*}{\mathcal {F}})}$

where

${\displaystyle {\begin{array}{rcl}X'&{\stackrel {g'}{\to }}&X\\f'\downarrow &&\downarrow f\\S'&{\stackrel {g}{\to }}&S\end{array}}}$

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.