Section 20.14 (01F7): Functoriality of cohomology

Lemma 20.14.1. Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}^\bullet$ , resp. $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_ Y$ -modules, resp. $\mathcal{O}_ X$ -modules. Let $\varphi : \mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet$ be a morphism of complexes. There is a canonical morphism

$\mathcal{G}^\bullet \longrightarrow Rf_*(\mathcal{F}^\bullet )$

in $D^{+}(Y)$ . Moreover this construction is functorial in the triple $(\mathcal{G}^\bullet , \mathcal{F}^\bullet , \varphi )$ .

Proof. Choose an injective resolution $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ . By definition $Rf_*(\mathcal{F}^\bullet )$ is represented by $f_*\mathcal{I}^\bullet$ in $K^{+}(\mathcal{O}_ Y)$ . The composition

$\mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet \to f_*\mathcal{I}^\bullet$

is a morphism in $K^{+}(Y)$ which turns into the morphism of the lemma upon applying the localization functor $j_ Y : K^{+}(Y) \to D^{+}(Y)$ . $\square$

Let

$f : X \to Y$ be a morphism of ringed spaces. Let

$\mathcal{G}$ be an

$\mathcal{O}_ Y$ -module and let

$\mathcal{F}$ be an

$\mathcal{O}_ X$ -module. Recall that an

$f$ -map

$\varphi$ from

$\mathcal{G}$ to

$\mathcal{F}$ is a map

$\varphi : \mathcal{G} \to f_*\mathcal{F}$ , or what is the same thing, a map

$\varphi : f^*\mathcal{G} \to \mathcal{F}$ . See Sheaves, Definition 6.21.7. Such an

$f$ -map gives rise to a morphism of complexes

$D^{+}(\mathcal{O}_ Y(Y))$ . Namely, we use the morphism

$\mathcal{G} \to Rf_*\mathcal{F}$ in

$D^{+}(Y)$ of Lemma 20.14.1, and we apply

$R\Gamma (Y, -)$ . By Lemma 20.13.1 we see that

$R\Gamma (X, \mathcal{F}) = R\Gamma (Y, Rf_*\mathcal{F})$ and we get the displayed arrow. We spell this out completely in Remark 20.14.2 below. In particular it gives rise to maps on cohomology

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

20.14 Functoriality of cohomology

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