The Stacks project

Lemma 37.56.10. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $(E, E', E'')$ be a distinguished triangle of $D(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$.

  1. If $E$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E'$ is $m$-pseudo-coherent relative to $S$ then $E''$ is $m$-pseudo-coherent relative to $S$.

  2. If $E, E''$ are $m$-pseudo-coherent relative to $S$, then $E'$ is $m$-pseudo-coherent relative to $S$.

  3. If $E'$ is $(m + 1)$-pseudo-coherent relative to $S$ and $E''$ is $m$-pseudo-coherent relative to $S$, then $E$ is $(m + 1)$-pseudo-coherent relative to $S$.

Moreover, if two out of three of $E, E', E''$ are pseudo-coherent relative to $S$, the so is the third.

Proof. Immediate from Lemma 37.56.6 and Cohomology, Lemma 20.44.4. $\square$


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