The Stacks project

102.8 Locally quasi-coherent modules with the flat base change property

Let $\mathcal{X}$ be an algebraic stack. We1 will denote

\[ \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \]

the full subcategory whose objects are étale $\mathcal{O}_\mathcal {X}$-modules $\mathcal{F}$ which are both locally quasi-coherent (Section 102.6) and have the flat base change property (Section 102.7). We have

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

by Sheaves on Stacks, Lemma 95.12.2.

Proposition 102.8.1. Summary of results on locally quasi-coherent modules having the flat base change property.

  1. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, then $\mathcal{F}$ is a sheaf for the fppf topology, i.e., it is an object of $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

  2. The category $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a weak Serre subcategory of both $\textit{Mod}(\mathcal{O}_\mathcal {X})$ and $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$.

  3. Pullback $f^*$ along any morphism of algebraic stacks $f : \mathcal{X} \to \mathcal{Y}$ induces a functor $f^* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

  4. If $f : \mathcal{X} \to \mathcal{Y}$ is a quasi-compact and quasi-separated morphism of algebraic stacks and $\mathcal{F}$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, then

    1. the derived direct image $Rf_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ can be computed in either the étale or the fppf topology with the same result, and

    2. each $R^ if_*\mathcal{F}$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$.

  5. The category $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ as well as in $\textit{Mod}(\mathcal{O}_\mathcal {X})$.

  6. Given $\mathcal{F}$ and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

  7. Given $\mathcal{F}$ of finite presentation and $\mathcal{G}$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Proof. Part (1) is Sheaves on Stacks, Lemma 95.23.1.

Part (2) for the embedding $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ we have seen in the proof of Lemma 102.7.3. Let us prove (2) for the embedding $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism between objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Since $\mathop{\mathrm{Ker}}(\varphi )$ is the same whether computed in the étale or the fppf topology, we see that $\mathop{\mathrm{Ker}}(\varphi )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ by the étale case. On the other hand, the cokernel computed in the fppf topology is the fppf sheafification of the cokernel computed in the étale topology. However, this étale cokernel is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ hence an fppf sheaf by (1) and we see that the cokernel is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Finally, suppose that

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

is an exact sequence in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ (i.e., using the fppf topology) with $\mathcal{F}_1$, $\mathcal{F}_2$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. In order to show that $\mathcal{F}_2$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ it suffices to show that the sequence is also exact in the étale topology. To do this it suffices to show that any element of $H^1_{fppf}(x, \mathcal{F}_1)$ becomes zero on the members of an étale covering of $x$ (for any object $x$ of $\mathcal{X}$). This is true because $H^1_{fppf}(x, \mathcal{F}_1) = H^1_{\acute{e}tale}(x, \mathcal{F}_1)$ by Sheaves on Stacks, Lemma 95.23.2 and because of locality of cohomology, see Cohomology on Sites, Lemma 21.7.3. This proves (2).

Part (3) follows from Lemma 102.7.2 and Sheaves on Stacks, Lemma 95.12.3.

Part (4)(b) for $R^ if_*\mathcal{F}$ computed in the étale cohomology follows from Lemma 102.7.3. Whereupon part (4)(a) follows from Sheaves on Stacks, Lemma 95.23.2 combined with (1) above.

Part (5) for the étale topology follows from Sheaves on Stacks, Lemma 95.12.4 and Lemma 102.7.2. The fppf version then follows as the colimit in the étale topology is already an fppf sheaf by part (1).

Parts (6) and (7) follow from the corresponding parts of Lemma 102.7.2 and Sheaves on Stacks, Lemma 95.12.4. $\square$

Lemma 102.8.2. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_ j|(|\mathcal{X}_ j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{\acute{e}tale}$. If each $f_ j^{-1}\mathcal{F}$ is in $\textit{LQCoh}^{fpc}(\mathcal{O}_{\mathcal{X}_ i})$, then $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

  2. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_ j|(|\mathcal{X}_ j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{fppf}$. If each $f_ j^{-1}\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_{\mathcal{X}_ i})$, then $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Proof. Part (1) follows from a combination of Lemmas 102.6.1 and 102.7.2. The proof of (2) is analogous to the proof of Lemma 102.6.3. Let $\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{fppf}$.

First, suppose there is a morphism $a : \mathcal{U} \to \mathcal{X}$ which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that $a^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Then there is an exact sequence

\[ 0 \to \mathcal{F} \to a_*a^*\mathcal{F} \to b_*b^*\mathcal{F} \]

where $b$ is the morphism $b : \mathcal{U} \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$, see Sheaves on Stacks, Proposition 95.19.7 and Lemma 95.19.10. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent and has the flat base change property, see Proposition 102.8.1. The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent and have the flat base change property by Proposition 102.8.1. We conclude that $\mathcal{F}$ is locally quasi-coherent and has the flat base change property by Proposition 102.8.1.

Choose a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By part (1) it suffices to show that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Again by part (1) it suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma 100.27.14 there exists an fppf covering $\{ a_ i : U_ i \to U\} $ such that each $x \circ a_ i$ factors through some $f_ j$. Hence the module $a_ i^*\mathcal{F}$ on $(\mathit{Sch}/U_ i)_{fppf}$ is locally quasi-coherent and has the flat base change property. After refining the covering we may assume $\{ U_ i \to U\} _{i = 1, \ldots , n}$ is a standard fppf covering. Then $x^*\mathcal{F}$ is an fppf module on $(\mathit{Sch}/U)_{fppf}$ whose pullback by the morphism $a : U_1 \amalg \ldots \amalg U_ n \to U$ is locally quasi-coherent and has the flat base change property. Hence by the previous paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property as desired. $\square$

Lemma 102.8.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks which is quasi-compact, quasi-separated, and representable by algebraic spaces. Let $\mathcal{F}$ be in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Then for an object $y : V \to \mathcal{Y}$ of $\mathcal{Y}$ we have

\[ (R^ if_*\mathcal{F})|_{V_{\acute{e}tale}} = R^ if'_{small, *}(\mathcal{F}|_{U_{\acute{e}tale}}) \]

where $f' : U = V \times _\mathcal {Y} \mathcal{X} \to V$ is the base change of $f$.

Proof. By Sheaves on Stacks, Lemma 95.21.3 we can reduce to the case where $\mathcal{X}$ is represented by $U$ and $\mathcal{Y}$ is represented by $V$. Of course this also uses that the pullback of $\mathcal{F}$ to $U$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_ U)$ by Proposition 102.8.1. Then the result follows from Sheaves on Stacks, Lemma 95.22.2 and the fact that $R^ if_*$ may be computed in the étale topology by Proposition 102.8.1. $\square$

Lemma 102.8.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be an affine morphism of algebraic stacks. The functor $f_* : \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \to \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {Y})$ is exact and commutes with direct sums. The functors $R^ if_*$ for $i > 0$ vanish on $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$.

Proof. The functors exist by Proposition 102.8.1. By Lemma 102.8.3 this reduces to the case of an affine morphism of algebraic spaces taking higher direct images in the setting of quasi-coherent modules on algebraic spaces. By the discussion in Cohomology of Spaces, Section 68.3 we reduce to the case of an affine morphism of schemes. For affine morphisms of schemes we have the vanishing of higher direct images on quasi-coherent modules by Cohomology of Schemes, Lemma 30.2.3. The vanishing for $R^1f_*$ implies exactness of $f_*$. Commuting with direct sums follows from Morphisms, Lemma 29.11.6 for example. $\square$

[1] Apologies for the horrendous notation.

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GQG. Beware of the difference between the letter 'O' and the digit '0'.