The Stacks project

104.6 Derived pushforward of quasi-coherent modules

As a first application of the material above we construct the derived pushforward. In Examples, Section 110.60 the reader can find an example of a quasi-compact and quasi-separated morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks such that the direct image functor $Rf_*$ does not induce a functor $D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to D_\mathit{QCoh}(\mathcal{O}_\mathcal {Y})$. Thus restricting to bounded below complexes is necessary.

Proposition 104.6.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $Rf_*$ induces a commutative diagram

\[ \xymatrix{ D^{+}_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[r] \ar[d]^{Rf_*} & D^{+}_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X}) \ar[r] \ar[d]^{Rf_*} & D(\mathcal{O}_\mathcal {X}) \ar[d]^{Rf_*} \\ D^{+}_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) \ar[r] & D^{+}_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y}) \ar[r] & D(\mathcal{O}_\mathcal {Y}) } \]

and hence induces a functor

\[ Rf_{\mathit{QCoh}, *} : D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow D^{+}_\mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \]

on quotient categories. Moreover, the functor $R^ if_\mathit{QCoh}$ of Cohomology of Stacks, Proposition 103.11.1 are equal to $H^ i \circ Rf_{\mathit{QCoh}, *}$ with $H^ i$ as in (104.5.1.1).

Proof. We have to show that $Rf_*E$ is an object of $D^{+}_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {Y})$ for $E$ in $D^{+}_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. This follows from Cohomology of Stacks, Proposition 103.8.1 and the spectral sequence $R^ if_*H^ j(E) \Rightarrow R^{i + j}f_*E$. The case of parasitic modules works the same way using Cohomology of Stacks, Lemma 103.9.3. The final statement is clear from the definition of $H^ i$ in (104.5.1.1). $\square$

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