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.