The Stacks project

Lemma 21.8.4. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. For any $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_\mathcal {C}))$ the sheaf $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

\[ V \longmapsto H^ i(u(V), \mathcal{F}) \]

Proof. Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then $R^ if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex

\[ f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots \]

By definition of the abelian category structure on $\mathcal{O}_\mathcal {D}$-modules this cohomology sheaf is the sheaf associated to the presheaf

\[ V \longmapsto \frac{\mathop{\mathrm{Ker}}(f_*\mathcal{I}^ i(V) \to f_*\mathcal{I}^{i + 1}(V))}{\mathop{\mathrm{Im}}(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^ i(V))} \]

and this is obviously equal to

\[ \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ i(u(V)) \to \mathcal{I}^{i + 1}(u(V)))}{\mathop{\mathrm{Im}}(\mathcal{I}^{i - 1}(u(V)) \to \mathcal{I}^ i(u(V)))} \]

which is equal to $H^ i(u(V), \mathcal{F})$ and we win. $\square$


Comments (2)

Comment #2169 by Kestutis Cesnavicius on

A tiny nitpick: a period is missing in the last sentence of the statement of the lemma.

Comment #2198 by on

OK, so I often allow myself to put in the period if the sentence ends with a displayed formula. My advisor Frans Oort always insisted I add the periods everywhere and of course one should. But right now this is such a low priority that I cannot bring myself to make changes like this. By all means somebody can go through the material and make style changes directly to the latex files and then send me the changes. IMPORTANT: Do a small sample first and send it to see if I am happy with the proposed changes.


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