The Stacks project

102.10 Quasi-coherent modules

We have seen that the category of quasi-coherent modules on an algebraic stack is equivalent to the category of quasi-coherent modules on a presentation, see Sheaves on Stacks, Section 95.15. This fact is the basis for the following.

Lemma 102.10.1. Let $\mathcal{X}$ be an algebraic stack. Let $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ be the category of locally quasi-coherent modules with the flat base change property, see Section 102.8. The inclusion functor $i : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ has a right adjoint

\[ Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

such that $Q \circ i$ is the identity functor.

Proof. Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Set $R = U \times _\mathcal {X} U$ so that we obtain a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces with the property that $\mathcal{X} = [U/R]$, see Algebraic Stacks, Lemma 93.16.2. We may and do replace $\mathcal{X}$ by $[U/R]$. By Sheaves on Stacks, Proposition 95.14.3 there is an equivalence

\[ q_1 : \mathit{QCoh}(U, R, s, t, c) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

Let us construct a functor

\[ q_2 : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(U, R, s, t, c) \]

by the following rule: if $\mathcal{F}$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then we set

\[ q_2(\mathcal{F}) = (f^*\mathcal{F}|_{U_{\acute{e}tale}}, \alpha ) \]

where $\alpha $ is the isomorphism

\[ t_{small}^*(f^*\mathcal{F}|_{U_{\acute{e}tale}}) \to t^*f^*\mathcal{F}|_{R_{\acute{e}tale}} \to s^*f^*\mathcal{F}|_{R_{\acute{e}tale}} \to s_{small}^*(f^*\mathcal{F}|_{U_{\acute{e}tale}}) \]

where the outer two morphisms are the comparison maps. Note that $q_2(\mathcal{F})$ is quasi-coherent precisely because $\mathcal{F}$ is locally quasi-coherent and that we used (and needed) the flat base change property in the construction of the descent datum $\alpha $. We omit the verification that the cocycle condition (see Groupoids in Spaces, Definition 77.12.1) holds. Looking at the proof of Sheaves on Stacks, Proposition 95.14.3 we see that $q_2 \circ i$ is the quasi-inverse to $q_1$. We define $Q = q_1 \circ q_2$. Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and let $\mathcal{G}$ be an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. We have

\begin{align*} \mathop{\mathrm{Mor}}\nolimits _{\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})} (i(\mathcal{G}), \mathcal{F}) & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(U, R, s, t, c)}(q_2(i(\mathcal{G})), q_2(\mathcal{F})) \\ & = \mathop{\mathrm{Mor}}\nolimits _{\mathit{QCoh}(\mathcal{O}_\mathcal {X})}(\mathcal{G}, Q(\mathcal{F})) \end{align*}

where the first equality is Sheaves on Stacks, Lemma 95.14.4 and the second equality holds because $q_1 \circ i$ and $q_2$ are quasi-inverse equivalences of categories. The assertion $Q \circ i \cong \text{id}$ is a formal consequence of the fact that $i$ is fully faithful. $\square$

Lemma 102.10.2. Let $\mathcal{X}$ be an algebraic stack. Let $Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$ be the functor constructed in Lemma 102.10.1.

  1. The kernel of $Q$ is exactly the collection of parasitic objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

  2. For any object $\mathcal{F}$ of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ both the kernel and the cokernel of the adjunction map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic.

  3. The functor $Q$ is exact and commutes with all limits and colimits.

Proof. Write $\mathcal{X} = [U/R]$ as in the proof of Lemma 102.10.1. Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. It is clear from the proof of Lemma 102.10.1 that $\mathcal{F}$ is in the kernel of $Q$ if and only if $\mathcal{F}|_{U_{\acute{e}tale}} = 0$. In particular, if $\mathcal{F}$ is parasitic then $\mathcal{F}$ is in the kernel. Next, let $x : V \to \mathcal{X}$ be a flat morphism, where $V$ is a scheme. Set $W = V \times _\mathcal {X} U$ and consider the diagram

\[ \xymatrix{ W \ar[d]_ p \ar[r]_ q & V \ar[d] \\ U \ar[r] & \mathcal{X} } \]

Note that the projection $p : W \to U$ is flat and the projection $q : W \to V$ is smooth and surjective. This implies that $q_{small}^*$ is a faithful functor on quasi-coherent modules. By assumption $\mathcal{F}$ has the flat base change property so that we obtain $p_{small}^*\mathcal{F}|_{U_{\acute{e}tale}} \cong q_{small}^*\mathcal{F}|_{V_{\acute{e}tale}}$. Thus if $\mathcal{F}$ is in the kernel of $Q$, then $\mathcal{F}|_{V_{\acute{e}tale}} = 0$ which completes the proof of (1).

Part (2) follows from the discussion above and the fact that the map $Q(\mathcal{F}) \to \mathcal{F}$ becomes an isomorphism after restricting to $U_{\acute{e}tale}$.

To see part (3) note that $Q$ is left exact as a right adjoint. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Consider the following commutative diagram

\[ \xymatrix{ 0 \ar[r] & Q(\mathcal{F}) \ar[r] \ar[d]_ a & Q(\mathcal{G}) \ar[r] \ar[d]_ b & Q(\mathcal{H}) \ar[r] \ar[d]_ c & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{G} \ar[r] & \mathcal{H} \ar[r] & 0 } \]

Since the kernels and cokernels of $a$, $b$, and $c$ are parasitic by part (2) and since the bottom row is a short exact sequence, we see that the top row as a complex of $\mathcal{O}_\mathcal {X}$-modules has parasitic cohomology sheaves (details omitted; this uses that the category of parasitic modules is a Serre subcategory of the category of all modules). By left exactness of $Q$ we see that only exactness at $Q(\mathcal{H})$ is at issue. However, the cokernel $\mathcal{Q}$ of $Q(\mathcal{G}) \to Q(\mathcal{H}))$ may be computed either in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ or in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with the same result because the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a left adjoint and hence right exact. Hence $\mathcal{Q} = Q(\mathcal{Q})$ is both quasi-coherent and parasitic, whence $0$ by part (1) as desired.

As a right adjoint $Q$ commutes with all limits. Since $Q$ is exact, to show that $Q$ commutes with all colimits it suffices to show that $Q$ commutes with direct sums, see Categories, Lemma 4.14.12. Let $\mathcal{F}_ i$, $i \in I$ be a family of objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. To see that $Q(\bigoplus \mathcal{F}_ i)$ is equal to $\bigoplus Q(\mathcal{F}_ i)$ we look at the construction of $Q$ in the proof of Lemma 102.10.1. This uses a presentation $\mathcal{X} = [U/R]$ where $U$ is a scheme. Then $Q(\mathcal{F})$ is computed by first taking the pair $(\mathcal{F}|_{U_{\acute{e}tale}}, \alpha )$ in $\mathit{QCoh}(U, R, s, t, c)$ and then using the equivalence $\mathit{QCoh}(U, R, s, t, c) \cong \mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Since the restriction functor $\textit{Mod}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_{U_{\acute{e}tale}})$, $\mathcal{F} \mapsto \mathcal{F}|_{U_{\acute{e}tale}}$ commutes with direct sums, the desired equality is clear. $\square$

Lemma 102.10.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Then $Q_\mathcal {X} \circ f^* = f^* \circ Q_\mathcal {Y}$ where $Q_\mathcal {X}$ and $Q_\mathcal {Y}$ are as in Lemma 102.10.1.

Proof. Observe that $f^*$ preserves both $\mathit{QCoh}$ and $\textit{LQCoh}^{fbc}$, see Sheaves on Stacks, Lemma 95.11.2 and Proposition 102.8.1. If $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ then $Q_\mathcal {Y}(\mathcal{F}) \to \mathcal{F}$ has parasitic kernel and cokernel by Lemma 102.10.2. As $f$ is flat we get that $f^*Q_\mathcal {Y}(\mathcal{F}) \to f^*\mathcal{F}$ has parasitic kernel and cokernel by Lemma 102.9.2. Thus the induced map $f^*Q_\mathcal {Y}(\mathcal{F}) \to Q_\mathcal {X}(f^*\mathcal{F})$ has parasitic kernel and cokernel and hence is an isomorphism for example by Lemma 102.9.4. $\square$

Lemma 102.10.4. Let $\mathcal{X}$ be an algebraic stack. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$ such that $x : U \to \mathcal{X}$ is flat. Then for $\mathcal{F}$ in $\mathit{QCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we have $Q(\mathcal{F})|_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}}$.

Proof. True because the kernel and cokernel of $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic, see Lemma 102.10.2. $\square$

Remark 102.10.5. Let $\mathcal{X}$ be an algebraic stack. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is abelian, the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ is right exact, but not exact in general, see Sheaves on Stacks, Lemma 95.15.1. We can use the functor $Q$ from Lemmas 102.10.1 and 102.10.2 to understand this. Namely, let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. Then

  1. the cokernel $\mathop{\mathrm{Coker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is quasi-coherent and is the cokernel of $\varphi $ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$,

  2. the image $\mathop{\mathrm{Im}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is quasi-coherent and is the image of $\varphi $ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$, and

  3. the kernel $\mathop{\mathrm{Ker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ by Proposition 102.8.1 and $Q(\mathop{\mathrm{Ker}}(\varphi ))$ is the kernel in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

This follows from the references given.

Remark 102.10.6. Let $\mathcal{X}$ be an algebraic stack. Given two quasi-coherent $\mathcal{O}_\mathcal {X}$-modules $\mathcal{F}$ and $\mathcal{G}$ the tensor product module $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ is quasi-coherent, see Sheaves on Stacks, Lemma 95.15.1 part (5). Similarly, given two locally quasi-coherent modules with the flat base change property, their tensor product has the same property, see Proposition 102.8.1. Thus the inclusion functors

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X}) \]

are functors of symmetric monoidal categories. What is more interesting is that the functor

\[ Q : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \longrightarrow \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

is a functor of symmetric monoidal categories as well. Namely, given $\mathcal{F}$ and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we obtain

\[ \xymatrix{ Q(\mathcal{F}) \otimes _{\mathcal{O}_\mathcal {X}} Q(\mathcal{G}) \ar[rr] \ar[rd] & & \mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G} \\ & Q(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}) \ar[ru] } \]

where the south-west arrow comes from the universal property of the north-west arrow (and the fact already mentioned that the object in the upper left corner is quasi-coherent). If we restrict this diagram to $U_{\acute{e}tale}$ for $U \to \mathcal{X}$ flat, then all three arrows become isomorphisms (see Lemmas 102.10.1 and 102.10.2 and Definition 102.9.1). Hence $Q(\mathcal{F}) \otimes _{\mathcal{O}_\mathcal {X}} Q(\mathcal{G}) \to Q(\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G})$ is an isomorphism, see for example Lemma 102.4.2.

Remark 102.10.7. Let $\mathcal{X}$ be an algebraic stack. Let $\textit{Parasitic}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$ denote the full subcategory consiting of parasitic modules. The results of Lemmas 102.10.1 and 102.10.2 imply that

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) / \textit{Parasitic}(\mathcal{O}_\mathcal {X}) \cap \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \]

in words: the category of quasi-coherent modules is the category of locally quasi-coherent modules with the flat base change property divided out by the Serre subcategory consisting of parasitic objects. See Homology, Lemma 12.10.6. The existence of the inclusion functor $i : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ which is left adjoint to the quotient functor is a key feature of the situation. In Derived Categories of Stacks, Section 103.5 and especially Lemma 103.5.4 we prove that a similar result holds on the level of derived categories.

Lemma 102.10.8. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module of finite presentation and let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_\mathcal {X}$-module. The internal homs $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{O}_\mathcal {X})$ agree and the common value is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. The quasi-coherent module $ hom(\mathcal{F}, \mathcal{G}) = Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) $ has the following universal property

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, hom(\mathcal{F}, \mathcal{G})) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{F}, \mathcal{G}) \]

for $\mathcal{H}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. The construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in Modules on Sites, Section 18.27 depends only on $\mathcal{F}$ and $\mathcal{G}$ as presheaves of modules; the output $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ is a sheaf for the fppf topology because $\mathcal{F}$ and $\mathcal{G}$ are assumed sheaves in the fppf topology, see Modules on Sites, Lemma 18.27.1. By Sheaves on Stacks, Lemma 95.12.4 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is locally quasi-coherent. By Lemma 102.7.2 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ has the flat base change property. Hence $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and it makes sense to apply the functor $Q$ of Lemma 102.10.1 to it. By the universal property of $Q$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}))) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) \]

for $\mathcal{H}$ quasi-coherent, hence the displayed formula of the lemma follows from Modules on Sites, Lemma 18.27.6. $\square$

Lemma 102.10.9. Let $f : \mathcal{X} \to \mathcal{Y}$ be a flat morphism of algebraic stacks. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {Y}$-module of finite presentation and let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_\mathcal {Y}$-module. Then $f^*hom(\mathcal{F}, \mathcal{G}) = hom(f^*\mathcal{F}, f^*\mathcal{G})$ with notation as in Lemma 102.10.8.

Proof. We have $f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {Y}}(\mathcal{F}, \mathcal{G}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(f^*\mathcal{F}, f^*\mathcal{G})$ by Modules on Sites, Lemma 18.31.4. (Observe that this step is not where the flatness of $f$ is used as the morphism of ringed topoi associated to $f$ is always flat, see Sheaves on Stacks, Remark 95.6.3.) Then apply Lemma 102.10.3 (and here we do use flatness of $f$). $\square$


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