The Stacks project

Lemma 20.43.4. Let $(X, \mathcal{O}_ X)$ be a ringed space and $m \in \mathbf{Z}$. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O}_ X)$.

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

  2. If $K$ and $M$ are $m$-pseudo-coherent, then $L$ is $m$-pseudo-coherent.

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

Proof. Proof of (1). Choose an open covering $X = \bigcup U_ i$ and maps $\alpha _ i : \mathcal{K}_ i^\bullet \to K|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $\mathcal{K}_ i^\bullet $ strictly perfect and $H^ j(\alpha _ i)$ isomorphisms for $j > m + 1$ and surjective for $j = m + 1$. We may replace $\mathcal{K}_ i^\bullet $ by $\sigma _{\geq m + 1}\mathcal{K}_ i^\bullet $ and hence we may assume that $\mathcal{K}_ i^ j = 0$ for $j < m + 1$. After refining the open covering we may choose maps $\beta _ i : \mathcal{L}_ i^\bullet \to L|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $\mathcal{L}_ i^\bullet $ strictly perfect such that $H^ j(\beta )$ is an isomorphism for $j > m$ and surjective for $j = m$. By Lemma 20.42.7 we can, after refining the covering, find maps of complexes $\gamma _ i : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $ such that the diagrams

\[ \xymatrix{ K|_{U_ i} \ar[r] & L|_{U_ i} \\ \mathcal{K}_ i^\bullet \ar[u]^{\alpha _ i} \ar[r]^{\gamma _ i} & \mathcal{L}_ i^\bullet \ar[u]_{\beta _ i} } \]

are commutative in $D(\mathcal{O}_{U_ i})$ (this requires representing the maps $\alpha _ i$, $\beta _ i$ and $K|_{U_ i} \to L|_{U_ i}$ by actual maps of complexes; some details omitted). The cone $C(\gamma _ i)^\bullet $ is strictly perfect (Lemma 20.42.2). The commutativity of the diagram implies that there exists a morphism of distinguished triangles

\[ (\mathcal{K}_ i^\bullet , \mathcal{L}_ i^\bullet , C(\gamma _ i)^\bullet ) \longrightarrow (K|_{U_ i}, L|_{U_ i}, M|_{U_ i}). \]

It follows from the induced map on long exact cohomology sequences and Homology, Lemmas 12.5.19 and 12.5.20 that $C(\gamma _ i)^\bullet \to M|_{U_ i}$ induces an isomorphism on cohomology in degrees $> m$ and a surjection in degree $m$. Hence $M$ is $m$-pseudo-coherent by Lemma 20.43.2.

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


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