The Stacks project

Lemma 109.60.1. Let $\mathcal{X}$ be an algebraic stack. Let $K$ be an object of $D(\mathcal{O}_\mathcal {X})$ whose cohomology sheaves are locally quasi-coherent (Sheaves on Stacks, Definition 95.12.1) and satisfy the flat base change property (Cohomology of Stacks, Definition 102.7.1). Then there exists a distinguished triangle

\[ K \to \prod \nolimits _{n \geq 0} \tau _{\geq -n} K \to \prod \nolimits _{n \geq 0} \tau _{\geq -n} K \to K[1] \]

in $D(\mathcal{O}_\mathcal {X})$. In other words, $K$ is the derived limit of its canonical truncations.

Proof. Recall that we work on the “big fppf site” $\mathcal{X}_{fppf}$ of $\mathcal{X}$ (by our conventions for sheaves of $\mathcal{O}_\mathcal {X}$-modules in the chapters Sheaves on Stacks and Cohomology on Stacks). Let $\mathcal{B}$ be the set of objects $x$ of $\mathcal{X}_{fppf}$ which lie over an affine scheme $U$. Combining Sheaves on Stacks, Lemmas 95.23.2, 95.16.1, Descent, Lemma 35.12.4, and Cohomology of Schemes, Lemma 30.2.2 we see that $H^ p(x, \mathcal{F}) = 0$ if $\mathcal{F}$ is locally quasi-coherent and $x \in \mathcal{B}$. Now the claim follows from Cohomology on Sites, Lemma 21.23.10 with $d = 0$. $\square$

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