5 fixed an error; added more detail

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 inverse direct images are exact when we 'change topology on the same category'category' (this includes $RX_{\ast EtNis}$ and $RX'_{\ast EtNis}$); hence the corresponding inverse 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?

4 Latex fix

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 $-{EtNis}$, -_{EtNis}$, then do we have $i^\ast{Nis}RX_{\ast i^\ast_{Nis}RX_{\ast EtNis}\cong RX'_{*EtNis}i^\ast_{Et}$? Is this true? It seems that I can prove it (using the fact that the inverse images are exact when we 'change topology on the same category'; hence the corresponding inverse 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? 3 added some details I have a closed embedding of schemes$X'\to i:X'\to X$, 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 inverse images are exact when we 'change topology on the same category'; hence the corresponding inverse 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?

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