The Stacks project

Lemma 35.9.5. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $T$. For either the étale or Zariski topology, there are canonical isomorphisms $R^ if_{small, *}(\mathcal{F}^ a) = (R^ if_*\mathcal{F})^ a$.

Proof. We prove this for the étale topology; we omit the proof in the case of the Zariski topology. By Cohomology of Schemes, Lemma 30.4.5 the sheaves $R^ if_*\mathcal{F}$ are quasi-coherent so that the assertion makes sense. The sheaf $R^ if_{small, *}\mathcal{F}^ a$ is the sheaf associated to the presheaf

\[ U \longmapsto H^ i(U \times _ S T, \mathcal{F}^ a) \]

where $g : U \to S$ is an object of $S_{\acute{e}tale}$, see Cohomology on Sites, Lemma 21.7.4. By our conventions the right hand side is the étale cohomology of the restriction of $\mathcal{F}^ a$ to the localization $T_{\acute{e}tale}/U \times _ S T$ which equals $(U \times _ S T)_{\acute{e}tale}$. By Proposition 35.9.3 this is presheaf the same as the presheaf

\[ U \longmapsto H^ i(U \times _ S T, (g')^*\mathcal{F}), \]

where $g' : U \times _ S T \to T$ is the projection. If $U$ is affine then this is the same as $H^0(U, R^ if'_*(g')^*\mathcal{F})$, see Cohomology of Schemes, Lemma 30.4.6. By Cohomology of Schemes, Lemma 30.5.2 this is equal to $H^0(U, g^*R^ if_*\mathcal{F})$ which is the value of $(R^ if_*\mathcal{F})^ a$ on $U$. Thus the values of the sheaves of modules $R^ if_{small, *}(\mathcal{F}^ a)$ and $(R^ if_*\mathcal{F})^ a$ on every affine object of $S_{\acute{e}tale}$ are canonically isomorphic which implies they are canonically isomorphic. $\square$

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