Definition 103.9.1. Let $\mathcal{X}$ be an algebraic stack. A presheaf of $\mathcal{O}_\mathcal {X}$-modules $\mathcal{F}$ is *parasitic* if we have $\mathcal{F}(x) = 0$ for any object $x$ of $\mathcal{X}$ which lies over a scheme $U$ such that the corresponding morphism $x : U \to \mathcal{X}$ is flat.

## 103.9 Parasitic modules

The following definition is compatible with Descent, Definition 35.12.1.

Here is a lemma with some properties of this notion.

Lemma 103.9.2. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_\mathcal {X}$-modules.

If $\mathcal{F}$ is parasitic and $g : \mathcal{Y} \to \mathcal{X}$ is a flat morphism of algebraic stacks, then $g^*\mathcal{F}$ is parasitic.

For $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $ we have

the $\tau $ sheafification of a parasitic presheaf of modules is parasitic, and

the full subcategory of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ consisting of parasitic modules is a Serre subcategory.

Suppose $\mathcal{F}$ is a sheaf for the étale topology. 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 $f_ i^*\mathcal{F}$ is parasitic then so is $\mathcal{F}$.

Suppose $\mathcal{F}$ is a sheaf for the fppf topology. Let $f_ i : \mathcal{X}_ i \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks such that $|\mathcal{X}| = \bigcup _ i |f_ i|(|\mathcal{X}_ i|)$. If each $f_ i^*\mathcal{F}$ is parasitic then so is $\mathcal{F}$.

**Proof.**
To see part (1) let $y$ be an object of $\mathcal{Y}$ which lies over a scheme $V$ such that the corresponding morphism $y : V \to \mathcal{Y}$ is flat. Then $g(y) : V \to \mathcal{Y} \to \mathcal{X}$ is flat as a composition of flat morphisms (see Morphisms of Stacks, Lemma 101.25.2) hence $\mathcal{F}(g(y))$ is zero by assumption. Since $g^*\mathcal{F} = g^{-1}\mathcal{F}(y) = \mathcal{F}(g(y))$ we conclude $g^*\mathcal{F}$ is parasitic.

To see part (2)(a) note that if $\{ x_ i \to x\} $ is a $\tau $-covering of $\mathcal{X}$, then each of the morphisms $x_ i \to x$ lies over a flat morphism of schemes. Hence if $x$ lies over a scheme $U$ such that $x : U \to \mathcal{X}$ is flat, so do all of the objects $x_ i$. Hence the presheaf $\mathcal{F}^+$ (see Sites, Section 7.10) is parasitic if the presheaf $\mathcal{F}$ is parasitic. This proves (2)(a) as the sheafification of $\mathcal{F}$ is $(\mathcal{F}^+)^+$.

Let $\mathcal{F}$ be a parasitic $\tau $-module. It is immediate from the definitions that any submodule of $\mathcal{F}$ is parasitic. On the other hand, if $\mathcal{F}' \subset \mathcal{F}$ is a submodule, then it is equally clear that the presheaf $x \mapsto \mathcal{F}(x)/\mathcal{F}'(x)$ is parasitic. Hence the quotient $\mathcal{F}/\mathcal{F}'$ is a parasitic module by (2)(a). Finally, we have to show that given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ parasitic, then $\mathcal{F}_2$ is parasitic. This follows immediately on evaluating on $x$ lying over a scheme flat over $\mathcal{X}$. This proves (2)(b), see Homology, Lemma 12.10.2.

Let $f_ i : \mathcal{X}_ i \to \mathcal{X}$ be a jointly surjective family of smooth morphisms of algebraic stacks and assume each $f_ i^*\mathcal{F}$ is parasitic. Let $x$ be an object of $\mathcal{X}$ which lies over a scheme $U$ such that $x : U \to \mathcal{X}$ is flat. Consider a surjective smooth covering $W_ i \to U \times _{x, \mathcal{X}} \mathcal{X}_ i$. Denote $y_ i : W_ i \to \mathcal{X}_ i$ the projection. It follows that $\{ f_ i(y_ i) \to x\} $ is a covering for the smooth topology on $\mathcal{X}$. Since a composition of flat morphisms is flat we see that $f_ i^*\mathcal{F}(y_ i) = 0$. On the other hand, as we saw in the proof of (1), we have $f_ i^*\mathcal{F}(y_ i) = \mathcal{F}(f_ i(y_ i))$. Hence we see that for some smooth covering $\{ x_ i \to x\} _{i \in I}$ in $\mathcal{X}$ we have $\mathcal{F}(x_ i) = 0$. This implies $\mathcal{F}(x) = 0$ because the smooth topology is the same as the étale topology, see More on Morphisms, Lemma 37.38.7. Namely, $\{ x_ i \to x\} _{i \in I}$ lies over a smooth covering $\{ U_ i \to U\} _{i \in I}$ of schemes. By the lemma just referenced there exists an étale covering $\{ V_ j \to U\} _{j \in J}$ which refines $\{ U_ i \to U\} _{i \in I}$. Denote $x'_ j = x|_{V_ j}$. Then $\{ x'_ j \to x\} $ is an étale covering in $\mathcal{X}$ refining $\{ x_ i \to x\} _{i \in I}$. This means the map $\mathcal{F}(x) \to \prod _{j \in J} \mathcal{F}(x'_ j)$, which is injective as $\mathcal{F}$ is a sheaf in the étale topology, factors through $\mathcal{F}(x) \to \prod _{i \in I} \mathcal{F}(x_ i)$ which is zero. Hence $\mathcal{F}(x) = 0$ as desired.

Proof of (4): omitted. Hint: similar, but simpler, than the proof of (3). $\square$

Parasitic modules are preserved under absolutely any pushforward.

Lemma 103.9.3. Let $\tau \in \{ {\acute{e}tale}, fppf\} $. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be a parasitic object of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$.

$H^ i_\tau (\mathcal{X}, \mathcal{F}) = 0$ for all $i$.

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Then $R^ if_*\mathcal{F}$ (computed in $\tau $-topology) is a parasitic object of $\textit{Mod}(\mathcal{Y}_\tau , \mathcal{O}_\mathcal {Y})$.

**Proof.**
We first reduce (2) to (1). By Sheaves on Stacks, Lemma 96.21.2 we see that $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

Here $y$ is a typical object of $\mathcal{Y}$ lying over the scheme $V$. By Lemma 103.9.2 it suffices to show that these cohomology groups are zero when $y : V \to \mathcal{Y}$ is flat. Note that $\text{pr} : V \times _{y, \mathcal{Y}} \mathcal{X} \to \mathcal{X}$ is flat as a base change of $y$. Hence by Lemma 103.9.2 we see that $\text{pr}^{-1}\mathcal{F}$ is parasitic. Thus it suffices to prove (1).

To see (1) we can use the spectral sequence of Sheaves on Stacks, Proposition 96.20.1 to reduce this to the case where $\mathcal{X}$ is an algebraic stack representable by an algebraic space. Note that in the spectral sequence each $f_ p^{-1}\mathcal{F} = f_ p^*\mathcal{F}$ is a parasitic module by Lemma 103.9.2 because the morphisms $f_ p : \mathcal{U}_ p = \mathcal{U} \times _\mathcal {X} \ldots \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$ are flat. Reusing this spectral sequence one more time (as in the proof of Lemma 103.5.1) we reduce to the case where the algebraic stack $\mathcal{X}$ is representable by a scheme $X$. Then $H^ i_\tau (\mathcal{X}, \mathcal{F}) = H^ i((\mathit{Sch}/X)_\tau , \mathcal{F})$. In this case the vanishing follows easily from an argument with Čech coverings, see Descent, Lemma 35.12.2. $\square$

The following lemma is one of the major reasons we care about parasitic modules. To understand the statement, recall that the functors $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ aren't exact in general.

Lemma 103.9.4. Let $\mathcal{X}$ be an algebraic stack. Let $\alpha : \mathcal{F} \to \mathcal{G}$ and $\beta : \mathcal{G} \to \mathcal{H}$ be maps in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\beta \circ \alpha = 0$. The following are equivalent:

in the abelian category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ the complex $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact at $\mathcal{G}$,

$\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ computed in either $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ is parasitic.

**Proof.**
We have $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \subset \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Section 103.8. Hence $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ agree, see Proposition 103.8.1. From now on we will use the étale topology on $\mathcal{X}$.

Let $\mathcal{E}$ be the cohomology of $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ computed in the abelian category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Let $x : U \to \mathcal{X}$ be a flat morphism where $U$ is a scheme. As we are using the étale topology, the restriction functor $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to \textit{Mod}(U_{\acute{e}tale}, \mathcal{O}_ U)$ is exact. On the other hand, by Lemma 103.4.1 and Sheaves on Stacks, Lemma 96.14.2 the restriction functor

is exact too. We conclude that $\mathcal{E}|_{U_{\acute{e}tale}} = (\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha ))|_{U_{\acute{e}tale}}$.

If (1) holds, then $\mathcal{E} = 0$ hence $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$ and this is the definition of a parasitic module. If (2) holds, then $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$ hence $\mathcal{E}$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$. This certainly implies that the quasi-coherent module $\mathcal{E}$ is zero, for example apply Lemma 103.4.2 to the map $0 \to \mathcal{E}$. $\square$

## 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.

## Comments (0)