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The Stacks project

Lemma 20.47.4. 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 20.46.7 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 20.46.2). 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 20.47.2.

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


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