The Stacks project

Lemma 22.35.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K^\bullet $ be a complex of $\mathcal{O}$-modules. Assume

  1. $K^\bullet $ represents a compact object of $D(\mathcal{O})$, and

  2. $E = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O})}(K^\bullet , K^\bullet )$ computes the ext groups of $K^\bullet $ in $D(\mathcal{O})$.

Then the functor

\[ - \otimes _ E^\mathbf {L} K^\bullet : D(E, \text{d}) \longrightarrow D(\mathcal{O}) \]

of Lemma 22.35.3 is fully faithful.

Proof. Because our functor has a left adjoint given by $R\mathop{\mathrm{Hom}}\nolimits (K^\bullet , -)$ by Lemma 22.35.5 it suffices to show for a differential graded $E$-module $M$ that the map

\[ H^0(M) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K^\bullet , M \otimes _ E^\mathbf {L} K^\bullet ) \]

is an isomorphism. We may assume that $M = P$ is a differential graded $E$-module which has property (P). Since $K^\bullet $ defines a compact object, we reduce using Lemma 22.20.1 to the case where $P$ has a finite filtration whose graded pieces are direct sums of $E[k]$. Again using compactness we reduce to the case $P = E[k]$. The assumption on $K^\bullet $ is that the result holds for these. $\square$

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