The Stacks project

Lemma 101.7.2. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module on $\mathcal{X}_{\acute{e}tale}$.

  1. If $\mathcal{F}$ has the flat base change property then for any morphism $g : \mathcal{Y} \to \mathcal{X}$ of algebraic stacks, the pullback $g^*\mathcal{F}$ does too.

  2. The full subcategory of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ consisting of modules with the flat base change property is a weak Serre subcategory.

  3. Let $f_ i : \mathcal{X}_ i \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks such that $|\mathcal{X}| = \bigcup _ i |f_ i|(|\mathcal{X}_ i|)$. If each $f_ i^*\mathcal{F}$ has the flat base change property then so does $\mathcal{F}$.

  4. The category of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{\acute{e}tale}$ with the flat base change property has colimits and they agree with colimits in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$.

Proof. Let $g : \mathcal{Y} \to \mathcal{X}$ be as in (1). Let $y$ be an object of $\mathcal{Y}$ lying over a scheme $V$. By Sheaves on Stacks, Lemma 94.9.3 we have $(g^*\mathcal{F})|_{V_{\acute{e}tale}} = \mathcal{F}|_{V_{\acute{e}tale}}$. Moreover a comparison mapping for the sheaf $g^*\mathcal{F}$ on $\mathcal{Y}$ is a special case of a comparison map for the sheaf $\mathcal{F}$ on $\mathcal{X}$, see Sheaves on Stacks, Lemma 94.9.3. In this way (1) is clear.

Proof of (2). We use the characterization of weak Serre subcategories of Homology, Lemma 12.10.3. Kernels and cokernels of maps between sheaves having the flat base change property also have the flat base change property. This is clear because $f_{small}^*$ is exact for a flat morphism of schemes and since the restriction functors $(-)|_{U_{\acute{e}tale}}$ are exact (because we are working in the étale topology). Finally, if $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and the outer two sheaves have the flat base change property then the middle one does as well, again because of the exactness of $f_{small}^*$ and the restriction functors (and the 5 lemma).

Proof of (3). Let $f_ i : \mathcal{X}_ i \to \mathcal{X}$ be a jointly surjective family of smooth morphisms of algebraic stacks and assume each $f_ i^*\mathcal{F}$ has the flat base change property. By part (1), the definition of an algebraic stack, and the fact that compositions of smooth morphisms are smooth (see Morphisms of Stacks, Lemma 99.32.2) we may assume that each $\mathcal{X}_ i$ is representable by a scheme. Let $\varphi : x \to x'$ be a morphism of $\mathcal{X}$ lying over a flat morphism $a : U \to U'$ of schemes. By Sheaves on Stacks, Lemma 94.18.10 there exists a jointly surjective family of étale morphisms $U'_ i \to U'$ such that $U'_ i \to U' \to \mathcal{X}$ factors through $\mathcal{X}_ i$. Thus we obtain commutative diagrams

\[ \xymatrix{ U_ i = U \times _{U'} U_ i' \ar[r]_-{a_ i} \ar[d] & U_ i' \ar[r]_{x_ i'} \ar[d] & \mathcal{X}_ i \ar[d]^{f_ i} \\ U \ar[r]^ a & U' \ar[r]^{x'} & \mathcal{X} } \]

Note that each $a_ i$ is a flat morphism of schemes as a base change of $a$. Denote $\psi _ i : x_ i \to x'_ i$ the morphism of $\mathcal{X}_ i$ lying over $a_ i$ with target $x_ i'$. By assumption the comparison maps $c_{\psi _ i} : (a_ i)_{small}^*\big (f_ i^*\mathcal{F}|_{(U'_ i)_{\acute{e}tale}}\big ) \to f_ i^*\mathcal{F}|_{(U_ i)_{\acute{e}tale}}$ is an isomorphism. Because the vertical arrows $U_ i' \to U'$ and $U_ i \to U$ are étale, the sheaves $f_ i^*\mathcal{F}|_{(U_ i')_{\acute{e}tale}}$ and $f_ i^*\mathcal{F}|_{(U_ i)_{\acute{e}tale}}$ are the restrictions of $\mathcal{F}|_{U'_{\acute{e}tale}}$ and $\mathcal{F}|_{U_{\acute{e}tale}}$ and the map $c_{\psi _ i}$ is the restriction of $c_\varphi $ to $(U_ i)_{\acute{e}tale}$, see Sheaves on Stacks, Lemma 94.9.3. Since $\{ U_ i \to U\} $ is an étale covering, this implies that the comparison map $c_\varphi $ is an isomorphism which is what we wanted to prove.

Proof of (4). Let $\mathcal{I} \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ be a diagram and assume each $\mathcal{F}_ i$ has the flat base change property. Recall that $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ is the sheafification of the presheaf colimit. As we are using the étale topology, it is clear that

\[ (\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)|_{U_{\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits _ i {\mathcal{F}_ i}|_{U_{\acute{e}tale}} \]

As $f_{small}^*$ commutes with colimits (as a left adjoint) we see that (4) holds. $\square$


Comments (2)

Comment #5138 by Dario Weißmann on

typo in the proof of (3) right above the diagram: U' \to U' \to curly X is missing an index i.


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