The Stacks project

102.13 Colimits and cohomology

The following lemma in particular applies to diagrams of quasi-coherent sheaves.

Lemma 102.13.1. Let $\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack. Then

\[ \mathop{\mathrm{colim}}\nolimits _ i H^ p(\mathcal{X}, \mathcal{F}_ i) \longrightarrow H^ p(\mathcal{X}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i) \]

is an isomorphism for every filtered diagram of abelian sheaves on $\mathcal{X}$. The same is true for abelian sheaves on $\mathcal{X}_{\acute{e}tale}$ taking cohomology in the étale topology.

Proof. Let $\tau = fppf$, resp. $\tau = {\acute{e}tale}$. The lemma follows from Cohomology on Sites, Lemma 21.16.2 applied to the site $\mathcal{X}_\tau $. In order to check the assumptions we use Cohomology on Sites, Remark 21.16.3. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_\tau )$ be the set of objects lying over affine schemes. In other words, an element of $\mathcal{B}$ is a morphism $x : U \to \mathcal{X}$ with $U$ affine. We check each of the conditions (1) – (4) of the remark in turn:

  1. Since $\mathcal{X}$ is quasi-compact, there exists a surjetive and smooth morphism $x : U \to \mathcal{X}$ with $U$ affine (Properties of Stacks, Lemma 99.6.2). Then $h_ x^\# \to *$ is a surjective map of sheaves on $\mathcal{X}_\tau $.

  2. Since coverings in $\mathcal{X}_\tau $ are fppf, resp. étale coverings, we see that every covering of $U \in \mathcal{B}$ is refined by a finite affine fppf covering, see Topologies, Lemma 34.7.4, resp. Lemma 34.4.4.

  3. Let $x : U \to \mathcal{X}$ and $x' : U' \to \mathcal{X}$ be in $\mathcal{B}$. The product $h_ x^\# \times h_{x'}^\# $ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_\tau )$ is equal to the sheaf on $\mathcal{X}_\tau $ determined by the algebraic space $W = U \times _{x, \mathcal{X}, x'} U'$ over $\mathcal{X}$: for an object $y : V \to \mathcal{X}$ of $\mathcal{X}_\tau $ we have $(h_ x^\# \times h_{x'}^\# )(y) = \{ f : V \to W \mid y = x \circ \text{pr}_1 \circ f = x' \circ \text{pr}_2 \circ f\} $. The algebraic space $W$ is quasi-compact because $\mathcal{X}$ is quasi-separated, see Morphisms of Stacks, Lemma 100.7.8 for example. Hence we can choose an affine scheme $U''$ and a surjective étale morphism $U'' \to W$. Denote $x'' : U'' \to \mathcal{X}$ the composition of $U'' \to W$ and $W \to \mathcal{X}$. Then $h_{x''}^\# \to h_ x^\# \times h_{x'}^\# $ is surjective as desired.

  4. Let $x : U \to \mathcal{X}$ and $x' : U' \to \mathcal{X}$ be in $\mathcal{B}$. Let $a, b : U \to U'$ be a morphism over $\mathcal{X}$, i.e., $a, b : x \to x'$ is a morphism in $\mathcal{X}_\tau $. Then the equalizer of $h_ a$ and $h_ b$ is represented by the equalizer of $a, b : U \to U'$ which is affine scheme over $\mathcal{X}$ and hence in $\mathcal{B}$.

This finished the proof. $\square$

Lemma 102.13.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be a filtered colimit of abelian sheaves on $\mathcal{X}$. Then for any $p \geq 0$ we have

\[ R^ pf_*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits R^ pf_*\mathcal{F}_ i. \]

The same is true for abelian sheaves on $\mathcal{X}_{\acute{e}tale}$ taking higher direct images in the étale topology.

Proof. We will prove this for the fppf topology; the proof for the étale topology is the same. Recall that $R^ if_*\mathcal{F}$ is the sheaf on $\mathcal{Y}_{fppf}$ associated to the presheaf

\[ (y : V \to \mathcal{Y}) \longmapsto H^ i(V \times _{y, \mathcal{Y}} \mathcal{X}, \text{pr}^{-1}\mathcal{F}) \]

See Sheaves on Stacks, Lemma 95.21.2. Recall that the colimit is the sheaf associated to the presheaf colimit. When $V$ is affine, the fibre product $V \times _\mathcal {Y} \mathcal{X}$ is quasi-compact and quasi-separated. Hence we can apply Lemma 102.13.1 to $H^ p(V \times _\mathcal {Y} \mathcal{X}, -)$ where $V$ is affine. Since every $V$ has an fppf covering by affine objects this proves the lemma. Some details omitted. $\square$

Lemma 102.13.3. Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $f_{\mathit{QCoh}, *}$ and the functors $R^ if_{\mathit{QCoh}, *}$ commute with direct sums and filtered colimits.

Proof. The functors $f_*$ and $R^ if_*$ commute with direct sums and filtered colimits on all modules by Lemma 102.13.2. The lemma follows as $f_{\mathit{QCoh}, *} = Q \circ f_*$ and $R^ if_{\mathit{QCoh}, *} = Q \circ R^ if_*$ and $Q$ commutes with all colimits, see Lemma 102.10.2. $\square$

Lemma 102.13.4. Let $f : \mathcal{X} \to \mathcal{Y}$ be an affine morphism of algebraic stacks. The functors $R^ if_{\mathit{QCoh}, *}$, $i > 0$ vanish and the functor $f_{\mathit{QCoh}, *}$ is exact and commutes with direct sums and all colimits.

Proof. Since we have $R^ if_{\mathit{QCoh}, *} = Q \circ R^ if_*$ we obtain the vanishing from Lemma 102.8.4. The vanishing implies that $f_{\mathit{QCoh}, *}$ is exact as $\{ R^ if_{\mathit{QCoh}, *}\} _{i \geq 0}$ form a $\delta $-functor, see Proposition 102.11.1. Then $f_{\mathit{QCoh}, *}$ commutes with direct sums for example by Lemma 102.13.3. An exact functor which commutes with direct sums commutes with all colimits. $\square$

The following lemma tells us that finitely presented modules behave as expected in quasi-compact and quasi-separated algebraic stacks.

Lemma 102.13.5. Let $\mathcal{X}$ be a quasi-compact and quasi-separated algebraic stack. Let $I$ be a directed set and let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system over $I$ of $\mathcal{O}_\mathcal {X}$-modules. Let $\mathcal{G}$ be an $\mathcal{O}_\mathcal {X}$-module of finite presentation. Then we have

\[ \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathcal{F}_ i) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i). \]

In particular, $\mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{G}, -)$ commutes with filtered colimits in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. The displayed equality is a special case of Modules on Sites, Lemma 18.27.12. In order to apply it, we need to check the hypotheses of Sites, Lemma 7.17.8 part (4) for the site $\mathcal{X}_{fppf}$. In order to do this, we will check hypotheses (2)(a), (2)(b), (2)(c) of Sites, Remark 7.17.9. Namely, let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{fppf})$ be the set of objects lying over affine schemes. In other words, an element of $\mathcal{B}$ is a morphism $x : U \to \mathcal{X}$ with $U$ affine. We check each of the conditions (2)(a), (2)(b), and (2)(c) of the remark in turn:

  1. Since $\mathcal{X}$ is quasi-compact, there exists a surjetive and smooth morphism $x : U \to \mathcal{X}$ with $U$ affine (Properties of Stacks, Lemma 99.6.2). Then $h_ x^\# \to *$ is a surjective map of sheaves on $\mathcal{X}_{fppf}$.

  2. Since coverings in $\mathcal{X}_{fppf}$ are fppf coverings, we see that every covering of $U \in \mathcal{B}$ is refined by a finite affine fppf covering, see Topologies, Lemma 34.7.4.

  3. Let $x : U \to \mathcal{X}$ and $x' : U' \to \mathcal{X}$ be in $\mathcal{B}$. The product $h_ x^\# \times h_{x'}^\# $ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ is equal to the sheaf on $\mathcal{X}_{fppf}$ determined by the algebraic space $W = U \times _{x, \mathcal{X}, x'} U'$ over $\mathcal{X}$: for an object $y : V \to \mathcal{X}$ of $\mathcal{X}_{fppf}$ we have $(h_ x^\# \times h_{x'}^\# )(y) = \{ f : V \to W \mid y = x \circ \text{pr}_1 \circ f = x' \circ \text{pr}_2 \circ f\} $. The algebraic space $W$ is quasi-compact because $\mathcal{X}$ is quasi-separated, see Morphisms of Stacks, Lemma 100.7.8 for example. Hence we can choose an affine scheme $U''$ and a surjective étale morphism $U'' \to W$. Denote $x'' : U'' \to \mathcal{X}$ the composition of $U'' \to W$ and $W \to \mathcal{X}$. Then $h_{x''}^\# \to h_ x^\# \times h_{x'}^\# $ is surjective as desired.

For the final statement, observe that the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ commutes with colimits and that finitely presented modules are quasi-coherent. See Sheaves on Stacks, Lemma 95.15.1. $\square$


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