103.11 Pushforward of quasi-coherent modules
Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Consider the pushforward
f_* : \textit{Mod}(\mathcal{O}_\mathcal {X}) \longrightarrow \textit{Mod}(\mathcal{O}_\mathcal {Y})
It turns out that this functor almost never preserves the subcategories of quasi-coherent sheaves. For example, consider the morphism of schemes
j : X = \mathbf{A}^2_ k \setminus \{ 0\} \longrightarrow \mathbf{A}^2_ k = Y.
Associated to this we have the corresponding morphism of algebraic stacks
f = j_{big} : \mathcal{X} = (\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/Y)_{fppf} = \mathcal{Y}
The pushforward f_*\mathcal{O}_\mathcal {X} of the structure sheaf has global sections k[x, y]. Hence if f_*\mathcal{O}_\mathcal {X} is quasi-coherent on \mathcal{Y} then we would have f_*\mathcal{O}_\mathcal {X} = \mathcal{O}_\mathcal {Y}. However, consider T = \mathop{\mathrm{Spec}}(k) \to \mathbf{A}^2_ k = Y mapping to 0. Then \Gamma (T, f_*\mathcal{O}_\mathcal {X}) = 0 because X \times _ Y T = \emptyset whereas \Gamma (T, \mathcal{O}_\mathcal {Y}) = k. On the positive side, for any flat morphism T \to Y we have the equality \Gamma (T, f_*\mathcal{O}_\mathcal {X}) = \Gamma (T, \mathcal{O}_\mathcal {Y}) as follows from Cohomology of Schemes, Lemma 30.5.2 using that j is quasi-compact and quasi-separated.
Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. We work around the problem mentioned above using the following three observations:
f_* does preserve locally quasi-coherent modules (Lemma 103.6.2),
f_* transforms a quasi-coherent sheaf into a locally quasi-coherent sheaf whose flat comparison maps are isomorphisms (Lemma 103.7.3), and
locally quasi-coherent \mathcal{O}_\mathcal {Y}-modules with the flat base change property give rise to quasi-coherent modules on a presentation of \mathcal{Y} and hence quasi-coherent modules on \mathcal{Y}, see Sheaves on Stacks, Section 96.15.
Thus we obtain a functor
f_{\mathit{QCoh}, *} : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
which is a right adjoint to f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X}) such that moreover
\Gamma (y, f_*\mathcal{F}) = \Gamma (y, f_{\mathit{QCoh}, *}\mathcal{F})
for any y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}) such that the associated 1-morphism y : V \to \mathcal{Y} is flat, see Lemma 103.11.2. Moreover, a similar construction will produce functors R^ if_{\mathit{QCoh}, *}. However, these results will not be sufficient to produce a total direct image functor (of complexes with quasi-coherent cohomology sheaves).
Proposition 103.11.1. Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X}) has a right adjoint
f_{\mathit{QCoh}, *} : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
which can be defined as the composition
\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \xrightarrow {f_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \xrightarrow {Q} \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
where the functors f_* and Q are as in Proposition 103.8.1 and Lemma 103.10.1. Moreover, if we define R^ if_{\mathit{QCoh}, *} as the composition
\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \xrightarrow {R^ if_*} \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \xrightarrow {Q} \mathit{QCoh}(\mathcal{O}_\mathcal {Y})
then the sequence of functors \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} forms a cohomological \delta -functor.
Proof.
This is a combination of the results mentioned in the statement. The adjointness can be shown as follows: Let \mathcal{F} be a quasi-coherent \mathcal{O}_\mathcal {X}-module and let \mathcal{G} be a quasi-coherent \mathcal{O}_\mathcal {Y}-module. Then we have
\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(f^*\mathcal{G}, \mathcal{F}) & = \mathop{\mathrm{Mor}}\nolimits _{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})} (\mathcal{G}, f_*\mathcal{F}) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(\mathcal{G}, Q(f_*\mathcal{F})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {Y})}(\mathcal{G}, f_{\mathit{QCoh}, *}\mathcal{F}) \end{align*}
the first equality by adjointness of f_* and f^* (for arbitrary sheaves of modules). By Proposition 103.8.1 we see that f_*\mathcal{F} is an object of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) (and can be computed in either the fppf or étale topology) and we obtain the second equality by Lemma 103.10.1. The third equality is the definition of f_{\mathit{QCoh}, *}.
To see that \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} is a cohomological \delta -functor as defined in Homology, Definition 12.12.1 let
0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0
be a short exact sequence of \mathit{QCoh}(\mathcal{O}_\mathcal {X}). This sequence may not be an exact sequence in \textit{Mod}(\mathcal{O}_\mathcal {X}) but we know that it is up to parasitic modules, see Lemma 103.9.4. Thus we may break up the sequence into short exact sequences
\begin{matrix} 0 \to \mathcal{P}_1 \to \mathcal{F}_1 \to \mathcal{I}_2 \to 0
\\ 0 \to \mathcal{I}_2 \to \mathcal{F}_2 \to \mathcal{Q}_2 \to 0
\\ 0 \to \mathcal{P}_2 \to \mathcal{Q}_2 \to \mathcal{I}_3 \to 0
\\ 0 \to \mathcal{I}_3 \to \mathcal{F}_3 \to \mathcal{P}_3 \to 0
\end{matrix}
of \textit{Mod}(\mathcal{O}_\mathcal {X}) with \mathcal{P}_ i parasitic. Note that each of the sheaves \mathcal{P}_ j, \mathcal{I}_ j, \mathcal{Q}_ j is an object of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}), see Proposition 103.8.1. Applying R^ if_* we obtain long exact sequences
\begin{matrix} 0 \to f_*\mathcal{P}_1 \to f_*\mathcal{F}_1 \to f_*\mathcal{I}_2 \to R^1f_*\mathcal{P}_1 \to \ldots
\\ 0 \to f_*\mathcal{I}_2 \to f_*\mathcal{F}_2 \to f_*\mathcal{Q}_2 \to R^1f_*\mathcal{I}_2 \to \ldots
\\ 0 \to f_*\mathcal{P}_2 \to f_*\mathcal{Q}_2 \to f_*\mathcal{I}_3 \to R^1f_*\mathcal{P}_2 \to \ldots
\\ 0 \to f_*\mathcal{I}_3 \to f_*\mathcal{F}_3 \to f_*\mathcal{P}_3 \to R^1f_*\mathcal{I}_3 \to \ldots
\end{matrix}
where are the terms are objects of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) by Proposition 103.8.1. By Lemma 103.9.3 the sheaves R^ if_*\mathcal{P}_ j are parasitic, hence vanish on applying the functor Q, see Lemma 103.10.2. Since Q is exact the maps
Q(R^ if_*\mathcal{F}_3) \cong Q(R^ if_*\mathcal{I}_3) \cong Q(R^ if_*\mathcal{Q}_2) \rightarrow Q(R^{i + 1}f_*\mathcal{I}_2) \cong Q(R^{i + 1}f_*\mathcal{F}_1)
can serve as the connecting map which turns the family of functors \{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0} into a cohomological \delta -functor.
\square
Lemma 103.11.2. Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. Let y : V \to \mathcal{Y} in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}) with y a flat morphism. Let \mathcal{F} be in \mathit{QCoh}(\mathcal{O}_\mathcal {X}). Then (f_*\mathcal{F})(y) = (f_{\mathit{QCoh}, *}\mathcal{F})(y) and (R^ if_*\mathcal{F})(y) = (R^ if_{\mathit{QCoh}, *}\mathcal{F})(y) for all i \in \mathbf{Z}.
Proof.
This follows from the construction of the functors R^ if_{\mathit{QCoh}, *} in Proposition 103.11.1, the definition of parasitic modules in Definition 103.9.1, and Lemma 103.10.2 part (2).
\square
Lemma 103.11.4. Let f : \mathcal{X} \to \mathcal{Y} be a quasi-compact and quasi-separated morphism of algebraic stacks. Let \mathcal{F} be a quasi-coherent sheaf on \mathcal{X}. Then there exists a spectral sequence with E_2-page
E_2^{p, q} = H^ p(\mathcal{Y}, R^ qf_{\mathit{QCoh}, *}\mathcal{F})
converging to H^{p + q}(\mathcal{X}, \mathcal{F}).
Proof.
By Cohomology on Sites, Lemma 21.14.5 the Leray spectral sequence with
E_2^{p, q} = H^ p(\mathcal{Y}, R^ qf_*\mathcal{F})
converges to H^{p + q}(\mathcal{X}, \mathcal{F}). The kernel and cokernel of the adjunction map
R^ qf_{\mathit{QCoh}, *}\mathcal{F} \longrightarrow R^ qf_*\mathcal{F}
are parasitic modules on \mathcal{Y} (Lemma 103.10.2) hence have vanishing cohomology (Lemma 103.9.3). It follows formally that H^ p(\mathcal{Y}, R^ qf_{\mathit{QCoh}, *}\mathcal{F}) = H^ p(\mathcal{Y}, R^ qf_*\mathcal{F}) and we win.
\square
Lemma 103.11.5. Let f : \mathcal{X} \to \mathcal{Y} and g : \mathcal{Y} \to \mathcal{Z} be quasi-compact and quasi-separated morphisms of algebraic stacks. Let \mathcal{F} be a quasi-coherent sheaf on \mathcal{X}. Then there exists a spectral sequence with E_2-page
E_2^{p, q} = R^ pg_{\mathit{QCoh}, *}(R^ qf_{\mathit{QCoh}, *}\mathcal{F})
converging to R^{p + q}(g \circ f)_{\mathit{QCoh}, *}\mathcal{F}.
Proof.
By Cohomology on Sites, Lemma 21.14.7 the Leray spectral sequence with
E_2^{p, q} = R^ pg_*(R^ qf_*\mathcal{F})
converges to R^{p + q}(g \circ f)_*\mathcal{F}. By the results of Proposition 103.8.1 all the terms of this spectral sequence are objects of \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Z}). Applying the exact functor Q_\mathcal {Z} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Z}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {Z}) we obtain a spectral sequence in \mathit{QCoh}(\mathcal{O}_\mathcal {Z}) covering to R^{p + q}(g \circ f)_{\mathit{QCoh}, *}\mathcal{F}. Hence the result follows if we can show that
Q_\mathcal {Z}(R^ pg_*(R^ qf_*\mathcal{F})) = Q_\mathcal {Z}(R^ pg_*(Q_\mathcal {X}(R^ qf_*\mathcal{F}))
This follows from the fact that the kernel and cokernel of the map
Q_\mathcal {X}(R^ qf_*\mathcal{F}) \longrightarrow R^ qf_*\mathcal{F}
are parasitic (Lemma 103.10.2) and that R^ pg_* transforms parasitic modules into parasitic modules (Lemma 103.9.3).
\square
To end this section we make explicit the spectral sequences associated to a smooth covering by a scheme. Please compare with Sheaves on Stacks, Sections 96.20 and 96.21.
Proposition 103.11.6. Let f : \mathcal{U} \to \mathcal{X} be a morphism of algebraic stacks. Assume f is representable by algebraic spaces, surjective, flat, and locally of finite presentation. Let \mathcal{F} be a quasi-coherent \mathcal{O}_\mathcal {X}-module. Then there is a spectral sequence
E_2^{p, q} = H^ q(\mathcal{U}_ p, f_ p^*\mathcal{F}) \Rightarrow H^{p + q}(\mathcal{X}, \mathcal{F})
where f_ p is the morphism \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X} (p + 1 factors).
Proof.
This is a special case of Sheaves on Stacks, Proposition 96.20.1.
\square
Proposition 103.11.7. Let f : \mathcal{U} \to \mathcal{X} and g : \mathcal{X} \to \mathcal{Y} be composable morphisms of algebraic stacks. Assume that
f is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and
g is quasi-compact and quasi-separated.
If \mathcal{F} is in \mathit{QCoh}(\mathcal{O}_\mathcal {X}) then there is a spectral sequence
E_2^{p, q} = R^ q(g \circ f_ p)_{\mathit{QCoh}, *}f_ p^*\mathcal{F} \Rightarrow R^{p + q}g_{\mathit{QCoh}, *}\mathcal{F}
in \mathit{QCoh}(\mathcal{O}_\mathcal {Y}).
Proof.
Note that each of the morphisms f_ p : \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X} is quasi-compact and quasi-separated, hence g \circ f_ p is quasi-compact and quasi-separated, hence the assertion makes sense (i.e., the functors R^ q(g \circ f_ p)_{\mathit{QCoh}, *} are defined). There is a spectral sequence
E_2^{p, q} = R^ q(g \circ f_ p)_*f_ p^{-1}\mathcal{F} \Rightarrow R^{p + q}g_*\mathcal{F}
by Sheaves on Stacks, Proposition 96.21.1. Applying the exact functor Q_\mathcal {Y} : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {Y}) gives the desired spectral sequence in \mathit{QCoh}(\mathcal{O}_\mathcal {Y}).
\square
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