Lemma 15.65.6. Let $R$ be a ring. Let $K \in D(R)$ be pseudo-coherent. Let $i \in \mathbf{Z}$. There exists a finitely presented $R$-module $M$ and a map $K \to M[-i]$ in $D(R)$ which induces an injection $H^ i(K) \to M$.

Proof. By Definition 15.64.1 we may represent $K$ by a complex $P^\bullet$ of finite free $R$-modules. Set $M = \mathop{\mathrm{Coker}}(P^{i - 1} \to P^ i)$. $\square$

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