I am reading M. Artin's treatment of the proper base change theorem for étale cohomology in his "Théorèms de représentabilité pur les espaces algébriques", and I have trouble understanding the following remark on page 222:

If $f:X\rightarrow S$ and $g:S'\rightarrow S$ are morphisms of algebraic spaces (or schemes, if you prefer), and if $f':X'\rightarrow S'$, $g':X'\rightarrow X$ denote the base changes of $f$ and $g$, then one can construct for any abelian sheaf $F$ on the big étale site of $X$ the base change morphism $g^*R^qf_*F\rightarrow R^q f'_*(g'^*F)$ (the higher direct images also computed on the big sites). If I understand correctly, Artin claims that if $F$, $R^q f_*F$ and $R^qf'_*(g'^*F)$ are representable on the big étale site of $X$, resp. $S$, resp. $S'$ (i.e. locally constructable), then the base change morphism is an isomorphism.

Why is that? Is that an easy fact?


1 Answer 1


With help from Milne's book on étale cohomology, I figured out how answer the question, although I am not sure that this argument is what Artin had in mind, and I still think that there's an easier argument.

There are morphisms of topoi $\pi_X: X_{ET}\rightarrow X_{et}$ from the topos associated to the big étale site of $X$ to the topos of the small étale site. Similarly for $S$, and $f:X\rightarrow S$ induces $f^s:X_{et}\rightarrow S_{et}$ and $f^b:X_{ET}\rightarrow S_{ET}$, and the obvious diagram commutes, i.e. $f^s\pi_X=\pi_Sf^b$. Given a sheaf $F$ in $S_{ET}$ we get a base change morphism $$ \pi_S^*R^qf^s_*F\rightarrow R^qf_*^b\pi_X^*F$$ Milne calls this "universal base change morphism", for good reasons: Given any morphism $g:S'\rightarrow S$, you also get morphisms of topoi $g^b:S'_{ET}\rightarrow S_{ET}$ and ${g'}^b:X'_{ET}\rightarrow X_{ET}$. Using this to restrict the universal base change morphism to $S'_{ET}$, we get the usual base change morphism for $g$ and $F$. (For this one has to check the commutativity of a few diagrams. All the ingredients can be found, e.g., in great detail in the Stacks Project)

Now, if $F$ is locally constructible, i.e. if the adjunction map $F\rightarrow \pi_X^*\pi_{X,*} F$ is an isomorphism, then it is not hard to check that the universal base change morphism is an isomorphism, and thus every base change morphism is an isorphism.


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