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The Stacks project

Lemma 103.13.1. 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 100.6.2). 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 101.7.8 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


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