The Stacks project

Lemma 20.43.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. If $K \oplus L$ is $m$-pseudo-coherent (resp. pseudo-coherent) in $D(\mathcal{O}_ X)$ so are $K$ and $L$.

Proof. Assume that $K \oplus L$ is $m$-pseudo-coherent. After replacing $X$ by the members of an open covering we may assume $K \oplus L \in D^-(\mathcal{O}_ X)$, hence $L \in D^-(\mathcal{O}_ X)$. Note that there is a distinguished triangle

\[ (K \oplus L, K \oplus L, L \oplus L[1]) = (K, K, 0) \oplus (L, L, L \oplus L[1]) \]

see Derived Categories, Lemma 13.4.10. By Lemma 20.43.4 we see that $L \oplus L[1]$ is $m$-pseudo-coherent. Hence also $L[1] \oplus L[2]$ is $m$-pseudo-coherent. By induction $L[n] \oplus L[n + 1]$ is $m$-pseudo-coherent. Since $L$ is bounded above we see that $L[n]$ is $m$-pseudo-coherent for large $n$. Hence working backwards, using the distinguished triangles

\[ (L[n], L[n] \oplus L[n - 1], L[n - 1]) \]

we conclude that $L[n - 1], L[n - 2], \ldots , L$ are $m$-pseudo-coherent as desired. $\square$


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