101.6 Locally quasi-coherent modules

Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal {X}$-modules. We can ask whether $\mathcal{F}$ is locally quasi-coherent, see Sheaves on Stacks, Definition 94.11.5. Briefly, this means $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module for the étale topology such that for any morphism $f : U \to \mathcal{X}$ the restriction $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent on $U_{\acute{e}tale}$. (The actual definition is slightly different, but equivalent.) A useful fact is that

$\textit{LQCoh}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$

is a weak Serre subcategory, see Sheaves on Stacks, Lemma 94.11.8.

Lemma 101.6.1. Let $\mathcal{X}$ be an algebraic stack. 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 locally quasi-coherent, then so is $\mathcal{F}$.

Proof. We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 99.33.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 94.11.7. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective smooth morphism. Let $x$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 94.18.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x$, $x_ i$ live over schemes $V$, $V_ i$, that $\{ V_ i \to V\}$ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. The restriction $x_ i^*\mathcal{F}|_{V_{i, {\acute{e}tale}}}$ is equal to the restriction of $f^*\mathcal{F}$ to $V_{i, {\acute{e}tale}}$, see Sheaves on Stacks, Lemma 94.9.3. Hence $x^*\mathcal{F}|_{V_{\acute{e}tale}}$ is a sheaf on the small étale site of $V$ which is quasi-coherent when restricted to $V_{i, {\acute{e}tale}}$ for each $i$. This implies that it is quasi-coherent (as desired), for example by Properties of Spaces, Lemma 64.29.6. $\square$

Lemma 101.6.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F}$ be a locally quasi-coherent $\mathcal{O}_\mathcal {X}$-module on $\mathcal{X}_{\acute{e}tale}$. Then $R^ if_*\mathcal{F}$ (computed in the étale topology) is locally quasi-coherent on $\mathcal{Y}_{\acute{e}tale}$.

Proof. We will use Lemma 101.5.1 to prove this. We will check its assumptions (1) – (4). Parts (1) and (2) follows from Sheaves on Stacks, Lemma 94.11.8. Part (3) follows from Lemma 101.6.1. Thus it suffices to show (4).

Suppose $f : \mathcal{X} \to \mathcal{Y}$ is a morphism of algebraic stacks such that $\mathcal{X}$ and $\mathcal{Y}$ are representable by affine schemes $X$ and $Y$. Choose any object $y$ of $\mathcal{Y}$ lying over a scheme $V$. For clarity, denote $\mathcal{V} = (\mathit{Sch}/V)_{fppf}$ the algebraic stack corresponding to $V$. Consider the cartesian diagram

$\xymatrix{ \mathcal{Z} \ar[d] \ar[r]_ g \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{V} \ar[r]^ y & \mathcal{Y} }$

Thus $\mathcal{Z}$ is representable by the scheme $Z = V \times _ Y X$ and $f'$ is quasi-compact and separated (even affine). By Sheaves on Stacks, Lemma 94.21.3 we have

$R^ if_*\mathcal{F}|_{V_{\acute{e}tale}} = R^ if'_{small, *}\big (g^*\mathcal{F}|_{Z_{\acute{e}tale}}\big )$

The right hand side is a quasi-coherent sheaf on $V_{\acute{e}tale}$ by Cohomology of Spaces, Lemma 67.3.1. This implies the left hand side is quasi-coherent which is what we had to prove. $\square$

Lemma 101.6.3. Let $\mathcal{X}$ be an algebraic stack. 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 locally quasi-coherent, then so is $\mathcal{F}$.

Proof. 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. 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 94.18.7 and Lemma 94.18.10. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent (Sheaves on Stacks, Lemma 94.11.7). The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent by Lemma 101.6.2. (Note that $a_*$ and $b_*$ don't care about which topology is used to calculate them.) We conclude that $\mathcal{F}$ is locally quasi-coherent, see Sheaves on Stacks, Lemma 94.11.8.

We are going to reduce the proof of the general case the situation in the first paragraph. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. We have to show that $\mathcal{F}|_{U_{\acute{e}tale}}$ is a quasi-coherent $\mathcal{O}_ U$-module. It suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma 99.27.14 there exists an fppf covering $\{ a_ i : U_ i \to U\}$ such that each $x \circ a_ i$ factors through some $f_ j$. Hence $a_ i^*\mathcal{F}$ is locally quasi-coherent on $(\mathit{Sch}/U_ i)_{fppf}$. 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. Hence by the first paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent, which certainly implies that $\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent. $\square$

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