The Stacks project

103.12 Further remarks on quasi-coherent modules

In this section we collect some results that to help understand how to use quasi-coherent modules on algebraic stacks.

Let $f : \mathcal{U} \to \mathcal{X}$ be a morphism of algebraic stacks. Assume $\mathcal{U}$ is represented by the algebraic space $U$. Consider the functor

given by pullback (Sheaves on Stacks, Section 96.7) followed by restriction (Sheaves on Stacks, Section 96.10). Applying this functor to locally quasi-coherent modules we obtain a functor

See Sheaves on Stacks, Lemmas 96.12.3 and 96.14.1. We can further limit our functor to even smaller subcategories to obtain

and

About these functors we can say the following:¹

1. The functor $a$ is exact. Namely, pullback $f^* = f^{-1}$ is exact (Sheaves on Stacks, Section 96.7) and restriction to $U_{\acute{e}tale}$ is exact, see Sheaves on Stacks, Equation (96.10.2.1).

2. The functor $b$ is exact. Namely, by Sheaves on Stacks, Lemma 96.12.4 the inclusion $\textit{LQCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is exact.

3. The functor $c$ is exact. Namely, by Proposition 103.8.1 the inclusion functor $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is exact.

4. The functor $d$ is right exact but not exact in general. Namely, by Sheaves on Stacks, Lemma 96.12.5 the inclusion functor $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ is right exact. We omit giving an example showing non-exactness.

5. If $f$ is flat, then $d$ is exact. This follows on combining Lemma 103.4.1 and Sheaves on Stacks, Lemma 96.14.2.

6. If $f$ is flat, then $c$ kills parasitic objects. Namely, $f^*$ preserves parasitic object by Lemma 103.9.2. Then for any scheme $V$ étale over $U$ and hence flat over $\mathcal{X}$ we see that $0 = f^*\mathcal{F}|_{V_{\acute{e}tale}} = c(\mathcal{F})|_{V_{\acute{e}tale}}$ by the compatibility of restriction with étale localization Sheaves on Stacks, Remark 96.10.2. Hence clearly $c(\mathcal{F}) = 0$.

7. If $f$ is flat, then $c = d \circ Q$. Namely, the kernel and cokernel of $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic by Lemma 103.10.2. Thus, since $c$ is exact (3) and kills parasitic objects (6), we see that $c$ applied to $Q(\mathcal{F}) \to \mathcal{F}$ is an isomorphism.

8. The functors $a, b, c, d$ commute with colimits and arbitrary direct sums. This is true for $f^*$ and restriction as left adjoints and hence it holds for $a$. Then it follows for $b$, $c$, $d$ by the references given above.

9. The functors $a, b, c, d$ commute with tensor products.

10. If $f$ is flat and surjective, $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, and $c(\mathcal{F}) = 0$, then $\mathcal{F}$ is parasitic. Namely, by (7) we get $d(Q(\mathcal{F})) = 0$. We may assume $U$ is a scheme by the compatibility of restriction with étale localization (see reference above). Then Lemma 103.4.2 applied to $0 \to Q(\mathcal{F})$ and the morphism $f : U \to \mathcal{X}$ shows that $Q(\mathcal{F}) = 0$. Thus $\mathcal{F}$ is parasitic by Lemma 103.10.2.

11. If $f$ is flat and surjective, then the functor $d$ reflects exactness. More precisely, let $\mathcal{F}^\bullet$ be a complex in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Then $\mathcal{F}^\bullet$ is exact in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ if and only if $d(\mathcal{F}^\bullet )$ is exact. Namely, we have seen one implication in (5). For the other, suppose that $H^ i(d(\mathcal{F}^\bullet )) = 0$. Then $\mathcal{G} = H^ i(\mathcal{F}^\bullet )$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $d(\mathcal{G}) = 0$. Hence $\mathcal{G}$ is both quasi-coherent and parasitic by (10), whence $0$ for example by Remark 103.10.7.

12. If $f$ is flat, $\mathcal{F}, \mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\mathit{QCoh}(\mathcal{O}_\mathcal {X}))$, and $\mathcal{F}$ of finite presentation and let then we have

    $$d(hom(\mathcal{F}, \mathcal{G})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(d(\mathcal{F}), d(\mathcal{G}))$$

    with notation as in Lemma 103.10.8. Perhaps the easiest way to see this is as follows

    $$\begin{align*} d(hom(\mathcal{F}, \mathcal{G})) & = d(Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}))) \\ & = c(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) \\ & = f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {U}}(f^*\mathcal{F}, f^*\mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(f^*\mathcal{F}|_{U_{\acute{e}tale}}, f^*\mathcal{G}|_{U_{\acute{e}tale}}) \end{align*}$$

    The first equality by construction of $hom$. The second equality by (7). The third equality by definition of $c$. The fourth equality by Modules on Sites, Lemma 18.31.4. The final equality by the same reference applied to the flat morphism of ringed topoi $i_ U (U_{\acute{e}tale}, \mathcal{O}_ U) \to (\mathcal{U}_{\acute{e}tale}, \mathcal{O}_\mathcal {U})$ of Sheaves on Stacks, Lemma 96.10.1.

13. add more here.