Lemma 22.35.3. The functor $K(\text{Mod}_{(E, \text{d})}) \to K(\mathcal{O})$ of Lemma 22.35.2 has a left derived version defined on all of $D(E, \text{d})$. We denote it $- \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \to D(\mathcal{O})$.
Proof. We will use Derived Categories, Lemma 13.14.15 to prove this. As our collection $\mathcal{P}$ of objects we will use the objects with property (P). Property (1) was shown in Lemma 22.20.4. Property (2) holds because if $s : P \to P'$ is a quasi-isomorphism of modules with property (P), then $s$ is a homotopy equivalence by Lemma 22.22.3. $\square$
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