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The Stacks project

Lemma 15.84.11. Let R be a ring. Let K_ j \in D(R), j = 1, 2, 3 with H^ i(K_ j) = 0 for i \not\in \{ -1, 0\} . Let \varphi : K_1 \to K_2 and \psi : K_2 \to K_3 be maps in D(R). If H^0(\varphi ) = 0 and H^{-1}(\psi ) = 0, then \varphi \circ \psi = 0.

Proof. Apply Derived Categories, Lemma 13.12.5 to see that \varphi \circ \psi factors through \tau _{\leq -2}K_2 = 0. \square


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