The Stacks project

102.11 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:

  1. $f_*$ does preserve locally quasi-coherent modules (Lemma 102.6.2),

  2. $f_*$ transforms a quasi-coherent sheaf into a locally quasi-coherent sheaf whose flat comparison maps are isomorphisms (Lemma 102.7.3), and

  3. 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 95.15.

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 102.11.2. 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 102.11.1. 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 102.8.1 and Lemma 102.10.1. 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 102.8.1 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 ├ętale topology) and we obtain the second equality by Lemma 102.10.1. 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 102.9.4. 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 102.8.1. 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 102.8.1. By Lemma 102.9.3 the sheaves $R^ if_*\mathcal{P}_ j$ are parasitic, hence vanish on applying the functor $Q$, see Lemma 102.10.2. 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 102.11.2. 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 102.11.1, the definition of parasitic modules in Definition 102.9.1, and Lemma 102.10.2 part (2). $\square$

Remark 102.11.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then there is a canonical commutative diagram

\[ \xymatrix{ f_{\mathit{QCoh}, *}\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_{\mathit{QCoh}, *}\mathcal{G} \ar[r] \ar[d] & f_*\mathcal{F} \otimes _{\mathcal{O}_\mathcal {Y}} f_*\mathcal{G} \ar[d]^ c \\ f_{\mathit{QCoh}, *}(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) \ar[r] & f_*(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) } \]

The vertical arrow $c$ on the right is the naive relative cup product (in degree $0$), see Cohomology on Sites, Section 21.33. The source and target of $c$ are in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Proposition 102.8.1. Applying $Q$ to $c$ we obtain the left vertical arrow as $Q$ commutes with tensor products, see Remark 102.10.6. This construction is functorial in $\mathcal{F}$ and $\mathcal{G}$.

Lemma 102.11.4. 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 102.10.2) hence have vanishing cohomology (Lemma 102.9.3). 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 102.11.5. 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 102.8.1 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 102.10.2) and that $R^ pg_*$ transforms parasitic modules into parasitic modules (Lemma 102.9.3). $\square$

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 95.20 and 95.21.

Proposition 102.11.6. 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 95.20.1. $\square$

Proposition 102.11.7. Let $f : \mathcal{U} \to \mathcal{X}$ and $g : \mathcal{X} \to \mathcal{Y}$ be composable morphisms of algebraic stacks. Assume that

  1. $f$ is representable by algebraic spaces, surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated, and

  2. $g$ 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 95.21.1. 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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