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

in $D^{+}(\mathcal{O}_ Y(Y))$. 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

Comments (0)

Post a comment