Remark 20.14.2 (01FB)—The Stacks project

Remark 20.14.2. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}$ be an $\mathcal{O}_ Y$-module. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Let $\varphi $ be an $f$-map from $\mathcal{G}$ to $\mathcal{F}$. Choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet $ by a complex of injective $\mathcal{O}_ X$-modules. Choose resolutions $\mathcal{G} \to \mathcal{J}^\bullet $ and $f_*\mathcal{I}^\bullet \to (\mathcal{J}')^\bullet $ by complexes of injective $\mathcal{O}_ Y$-modules. By Derived Categories, Lemma 13.18.6 there exists a map of complexes $\beta $ such that the diagram

20.14.2.1

\begin{equation} \label{cohomology-equation-choice} \xymatrix{ \mathcal{G} \ar[d] \ar[r] & f_*\mathcal{F} \ar[r] & f_*\mathcal{I}^\bullet \ar[d] \\ \mathcal{J}^\bullet \ar[rr]^\beta & & (\mathcal{J}')^\bullet } \end{equation}

commutes. Applying global section functors we see that we get a diagram

\[ \xymatrix{ & & \Gamma (Y, f_*\mathcal{I}^\bullet ) \ar[d]_{qis} \ar@{=}[r] & \Gamma (X, \mathcal{I}^\bullet ) \\ \Gamma (Y, \mathcal{J}^\bullet ) \ar[rr]^\beta & & \Gamma (Y, (\mathcal{J}')^\bullet ) & } \]

The complex on the bottom left represents $R\Gamma (Y, \mathcal{G})$ and the complex on the top right represents $R\Gamma (X, \mathcal{F})$. The vertical arrow is a quasi-isomorphism by Lemma 20.13.1 which becomes invertible after applying the localization functor $K^{+}(\mathcal{O}_ Y(Y)) \to D^{+}(\mathcal{O}_ Y(Y))$. The arrow (20.14.1.1) is given by the composition of the horizontal map by the inverse of the vertical map.

The Stacks project

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