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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)