I have a closed embedding of schemes $i:X'\to X$, and for each of them I consider three Grothendieck topologies for the category of the corresponding (relatively) \'etale schemes: the \'etale one, the Nisnevich one, and the one that I call 'trivial'. The latter topology is the one whose only covers are isomorphisms; so the sheaves wrt this topology are just presheaves.

Now, I have seven morphisms of sites in my situation, and I would like to say that I have base change (i.e. that all the higher direct image functors for sheaves 'commute' with the corresponding inverse image functors). For example, if we denote the morphisms of the corresponding sites by $\-_{EtNis}$, then do we have $i^\ast_{Nis}RX_{\ast EtNis}\cong RX'_{*EtNis}i^\ast_{Et}$?

Is this true? It seems that I can prove it (using the fact that the direct images are exact when we 'change topology on the same category' (this includes $RX_{\ast EtNis}$ and $RX'_{\ast EtNis}$); hence the corresponding direct images respect injective objects), and the proof is rather simple. Yet I am not quite sure that I am right, and I would definitely prefer to put a reference to this fact in my paper instead of proving it by myself. This should be an easy basic result on the base change (if true), but I do not know where to look for statements of this sort.

Also, is there a 'classical' name for the topology that I call the trivial one?

  • 2
    $\begingroup$ Sorry, but there are too many ways to interpret your question. Can you write exactly which morphisms of sites you are considering, and which direct image functors should commute with which inverse image functors ? By the way, you "trivial topology" is classically called "chaotic" or "coarse" (cf SGA 4 II 1.1.4). $\endgroup$ – Alex Oct 10 '11 at 19:03
  • $\begingroup$ Mikhail -- I've fixed the Latex. It seems that Mathjacks goes gaga once it sees an asterisk. $\endgroup$ – algori Oct 10 '11 at 20:30
  • $\begingroup$ I don't think that this can be true for general reasons. You write "It seems that I can prove it (using the fact that the inverse images are exact when we 'change topology on the same category'; hence the corresponding inverse images respect injective objects)", which doesn't make sense to me because there are too many inverse image functors in that sentence. Anyway, inverse image functors are always exact, so I'm going to assume you mean "direct images" the first time, and then I think that your sentence is not true (it seems that it would imply that the global sections functor is exact). $\endgroup$ – Alex Oct 10 '11 at 21:24
  • $\begingroup$ Dear Alex, thanks! I fixed direct/inverse images, and will think about your words. $\endgroup$ – Mikhail Bondarko Oct 11 '11 at 3:27

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