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Lars
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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 \[ \pi_S^*R^qf^s_*F\rightarrow R^qf_*^b\pi_X^*F\]$$ \pi_S^*R^qf^s_*F\rightarrow R^qf_*^b\pi_X^*F$$ 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.

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.

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.

Source Link
Lars
Lars
  • 4.5k
  • 3
  • 35
  • 48

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.