The Stacks project

Lemma 15.81.6. Let $R$ be a ring. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ be a distinguished triangle in $D(A)$.

  1. If $K^\bullet $ is $(m + 1)$-pseudo-coherent relative to $R$ and $L^\bullet $ is $m$-pseudo-coherent relative to $R$ then $M^\bullet $ is $m$-pseudo-coherent relative to $R$.

  2. If $K^\bullet , M^\bullet $ are $m$-pseudo-coherent relative to $R$, then $L^\bullet $ is $m$-pseudo-coherent relative to $R$.

  3. If $L^\bullet $ is $(m + 1)$-pseudo-coherent relative to $R$ and $M^\bullet $ is $m$-pseudo-coherent relative to $R$, then $K^\bullet $ is $(m + 1)$-pseudo-coherent relative to $R$.

Moreover, if two out of three of $K^\bullet , L^\bullet , M^\bullet $ are pseudo-coherent relative to $R$, the so is the third.

Proof. Follows immediately from Lemma 15.64.2 and the definitions. $\square$

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