The Stacks project

103.17 Coherent sheaves on locally Noetherian stacks

This section is the analogue of Cohomology of Spaces, Section 69.12. We have defined the notion of a coherent module on any ringed topos in Modules on Sites, Section 18.23. However, for any algebraic stack $\mathcal{X}$ the category of coherent $\mathcal{O}_\mathcal {X}$-modules is zero, essentially because the site $\mathcal{X}$ contains too many non-Noetherian objects (even if $\mathcal{X}$ is itself locally Noetherian). Instead, we will define coherent modules using the following lemma.

Lemma 103.17.1. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module. The following are equivalent

  1. $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_\mathcal {X}$-module,

  2. $\mathcal{F}$ is an $\mathcal{O}_\mathcal {X}$-module of finite presentation,

  3. $\mathcal{F}$ is quasi-coherent and for any morphism $f : U \to \mathcal{X}$ where $U$ is a locally Noetherian algebraic space, the pullback $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is coherent, and

  4. $\mathcal{F}$ is quasi-coherent and there exists an algebraic space $U$ and a morphism $f : U \to \mathcal{X}$ which is locally of finite type, flat, and surjective, such that the pullback $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is coherent.

Proof. Let $f : U \to \mathcal{X}$ be as in (4). Then $U$ is locally Noetherian (Morphisms of Stacks, Lemma 101.17.5) and we see that the statement of the lemma makes sense. Additionally, $f$ is locally of finite presentation by Morphisms of Stacks, Lemma 101.27.5. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $V$. In order to prove (2) we have to show that, after replacing $V$ by the members of an fppf covering of $V$, the restriction $x^*\mathcal{F}$ has a global finite presentation on $\mathcal{X}/x \cong (\mathit{Sch}/V)_{fppf}$. The projection $W = U \times _\mathcal {X} V \to V$ is locally of finite presentation, flat, and surjective. Hence we may replace $V$ by the members of an étale covering of $W$ by schemes and assume we have a morphism $h : V \to U$ with $f \circ h = x$. Since $\mathcal{F}$ is quasi-coherent, we see that the restriction $x^*\mathcal{F}$ is the pullback of $h_{small}^*(f^*\mathcal{F})|_{U_{\acute{e}tale}}$ by $\pi _ V$, see Sheaves on Stacks, Lemma 96.14.2. Since $f^*\mathcal{F}|_{U_{\acute{e}tale}}$ locally in the étale topology has a finite presentation by assumption, we conclude (4) $\Rightarrow $ (2).

Part (2) implies (1) for any ringed topos (immediate from the definition). The properties “finite type” and “quasi-coherent” are preserved under pullback by any morphism of ringed topoi, see Modules on Sites, Lemma 18.23.4. Hence (1) implies (3), see Cohomology of Spaces, Lemma 69.12.2. Finally, (3) trivially implies (4). $\square$

Definition 103.17.2. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. An $\mathcal{O}_\mathcal {X}$-module $\mathcal{F}$ is called coherent if $\mathcal{F}$ satisfies one (and hence all) of the equivalent conditions of Lemma 103.17.1. The category of coherent $\mathcal{O}_\mathcal {X}$-modules is denote $\textit{Coh}(\mathcal{O}_\mathcal {X})$.

Lemma 103.17.3. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. The module $\mathcal{O}_\mathcal {X}$ is coherent, any invertible $\mathcal{O}_\mathcal {X}$-module is coherent, and more generally any finite locally free $\mathcal{O}_\mathcal {X}$-module is coherent.

Proof. Follows from the definition and Cohomology of Spaces, Lemma 69.12.2. $\square$

Lemma 103.17.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of locally Noetherian algebraic stacks. Then $f^*$ sends coherent modules on $\mathcal{Y}$ to coherent modules on $\mathcal{X}$.

Proof. Immediate from the definition and the fact that pullback for any morphism of ringed topoi preserves finitely presented modules, see Modules on Sites, Lemma 18.23.4. $\square$

Lemma 103.17.5. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. The category of coherent $\mathcal{O}_\mathcal {X}$-modules is abelian. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of coherent $\mathcal{O}_\mathcal {X}$-modules, then

  1. the cokernel $\mathop{\mathrm{Coker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is a coherent $\mathcal{O}_\mathcal {X}$-module,

  2. the image $\mathop{\mathrm{Im}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is a coherent $\mathcal{O}_\mathcal {X}$-module, and

  3. the kernel $\mathop{\mathrm{Ker}}(\varphi )$ computed in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ may not be coherent, but it is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and $Q(\mathop{\mathrm{Ker}}(\varphi ))$ is coherent and is the kernel of $\varphi $ in $\textit{Coh}(\mathcal{O}_\mathcal {X})$.

The inclusion functor $\textit{Coh}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is exact.

Proof. The rules given for taking kernels, images, and cokernels in $\textit{Coh}(\mathcal{O}_\mathcal {X})$ agree with the prescription for quasi-coherent modules in Remark 103.10.5. Hence the lemma will follow if we can show that the quasi-coherent modules $\mathop{\mathrm{Coker}}(\varphi )$, $\mathop{\mathrm{Im}}(\varphi )$, and $Q(\mathop{\mathrm{Ker}}(\varphi ))$ are coherent. By Lemma 103.17.1 it suffices to prove this after restricting to $U_{\acute{e}tale}$ for some surjective smooth morphism $f : U \to \mathcal{X}$. The functor $\mathcal{F} \mapsto f^*\mathcal{F}|_{U_{\acute{e}tale}}$ is exact. Hence $f^*\mathop{\mathrm{Coker}}(\varphi )$ and $f^*\mathop{\mathrm{Im}}(\varphi )$ are the cokernel and image of a map between coherent $\mathcal{O}_ U$-modules hence coherent as desired. The functor $\mathcal{F} \mapsto f^*\mathcal{F}|_{U_{\acute{e}tale}}$ kills parasitic modules by Lemma 103.9.2. Hence $f^*Q(\mathop{\mathrm{Ker}}(\varphi ))|_{U_{\acute{e}tale}} = f^*\mathop{\mathrm{Ker}}(\varphi )|_{U_{\acute{e}tale}}$ by part (2) of Lemma 103.10.2. Thus we conclude that $Q(\mathop{\mathrm{Ker}}(\varphi ))$ is coherent in the same way. $\square$

Lemma 103.17.6. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ coherent, then $\mathcal{F}_2$ is coherent.

Proof. By Sheaves on Stacks, Lemma 96.15.1 part (7) we see that $\mathcal{F}_2$ is quasi-coherent. Then we can check that $\mathcal{F}_2$ is coherent by restricting to $U_{\acute{e}tale}$ for some $U \to \mathcal{X}$ surjective and smooth. This follows from Cohomology of Spaces, Lemma 69.12.3. Some details omitted. $\square$

Coherent modules form a Serre subcategory of the category of quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. This does not hold for modules on a general ringed topos.

Lemma 103.17.7. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Then $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. We have

  1. if $\mathcal{F}$ is coherent and $\varphi $ surjective, then $\mathcal{G}$ is coherent,

  2. if $\mathcal{F}$ is coherent, then $\mathop{\mathrm{Im}}(\varphi )$ is coherent, and

  3. if $\mathcal{G}$ coherent and $\mathop{\mathrm{Ker}}(\varphi )$ parasitic, then $\mathcal{F}$ is coherent.

Proof. Choose a scheme $U$ and a surjective smooth morphism $f : U \to \mathcal{X}$. Then the functor $f^* : \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \to \mathit{QCoh}(\mathcal{O}_ U)$ is exact (Lemma 103.4.1) and moreover by definition $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is the full subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ consisting of objects $\mathcal{F}$ such that $f^*\mathcal{F}$ is in $\textit{Coh}(\mathcal{O}_ U)$. The statement that $\textit{Coh}(\mathcal{O}_\mathcal {X})$ is a Serre subcategory of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ follows immediately from this and the corresponding fact for $U$, see Cohomology of Spaces, Lemmas 69.12.3 and 69.12.4. We omit the proof of (1), (2), and (3). Hint: compare with the proof of Lemma 103.17.5. $\square$

Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $U$ be an algebraic space and let $f : U \to \mathcal{X}$ be surjective, locally of finite presentation, and flat. Observe that $U$ is locally Noetherian (Morphisms of Stacks, Lemma 101.17.5). Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces and $f_{can} : [U/R] \to \mathcal{X}$ the isomorphism constructed in Algebraic Stacks, Lemma 94.16.1 and Remark 94.16.3. As in Sheaves on Stacks, Section 96.15 we obtain equivalences

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(\mathcal{O}_{[U/R]}) \cong \mathit{QCoh}(U, R, s, t, c) \]

where the second equivalence is Sheaves on Stacks, Proposition 96.14.3. Recall that in Groupoids in Spaces, Section 78.13 we have defined the full subcategory

\[ \textit{Coh}(U, R, s, t, c) \subset \mathit{QCoh}(U, R, s, t, c) \]

of coherent modules as those $(\mathcal{G}, \alpha )$ such that $\mathcal{G}$ is a coherent $\mathcal{O}_ U$-module.

Lemma 103.17.8. In the situation discussed above, the equivalence $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \cong \mathit{QCoh}(U, R, s, t, c)$ sends coherent sheaves to coherent sheaves and vice versa, i.e., induces an equivalence $\textit{Coh}(\mathcal{O}_\mathcal {X}) \cong \textit{Coh}(U, R, s, t, c)$.

Proof. This is immediate from the definition of coherent $\mathcal{O}_\mathcal {X}$-modules. For bookkeeping purposes: the material above uses Morphisms of Stacks, Lemma 101.17.5, Algebraic Stacks, Lemma 94.16.1 and Remark 94.16.3, Sheaves on Stacks, Section 96.15, Sheaves on Stacks, Proposition 96.14.3, and Groupoids in Spaces, Section 78.13. $\square$

Lemma 103.17.9. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $\mathcal{F}$ and $\mathcal{G}$ be coherent be $\mathcal{O}_\mathcal {X}$-modules. Then the internal hom $hom(\mathcal{F}, \mathcal{G})$ constructed in Lemma 103.10.8 is a coherent $\mathcal{O}_\mathcal {X}$-module.

Proof. Let $U \to \mathcal{X}$ be a smooth surjective morphism from a scheme. By item (12) in Section 103.12 we see that the restriction of $hom(\mathcal{F}, \mathcal{G})$ to $U$ is the Hom sheaf of the restrictions. Hence this lemma follows from the case of algebraic spaces, see Cohomology of Spaces, Lemma 69.12.5. $\square$

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