The Stacks project

Lemma 15.62.2. Let $R$ be a ring and $m \in \mathbf{Z}$. Let $(K^\bullet , L^\bullet , M^\bullet , f, g, h)$ be a distinguished triangle in $D(R)$.

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

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

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

Proof. Proof of (1). Choose $\alpha : P^\bullet \to K^\bullet $ with $P^\bullet $ a bounded complex of finite free modules such that $H^ i(\alpha )$ is an isomorphism for $i > m + 1$ and surjective for $i = m + 1$. We may replace $P^\bullet $ by $\sigma _{\geq m + 1}P^\bullet $ and hence we may assume that $P^ i = 0$ for $i < m + 1$. Choose $\beta : E^\bullet \to L^\bullet $ with $E^\bullet $ a bounded complex of finite free modules such that $H^ i(\beta )$ is an isomorphism for $i > m$ and surjective for $i = m$. By Derived Categories, Lemma 13.19.11 we can find a map $\alpha : P^\bullet \to E^\bullet $ such that the diagram

\[ \xymatrix{ K^\bullet \ar[r] & L^\bullet \\ P^\bullet \ar[u] \ar[r]^\alpha & E^\bullet \ar[u] } \]

is commutative in $D(R)$. The cone $C(\alpha )^\bullet $ is a bounded complex of finite free $R$-modules, and the commutativity of the diagram implies that there exists a morphism of distinguished triangles

\[ (P^\bullet , E^\bullet , C(\alpha )^\bullet ) \longrightarrow (K^\bullet , L^\bullet , M^\bullet ). \]

It follows from the induced map on long exact cohomology sequences and Homology, Lemmas 12.5.19 and 12.5.20 that $C(\alpha )^\bullet \to M^\bullet $ induces an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. Hence $M^\bullet $ is $m$-pseudo-coherent.

Assertions (2) and (3) follow from (1) by rotating the distinguished triangle. $\square$


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