Lemma 22.35.6 (09LZ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.35: Variant of derived tensor product
Lemma 22.35.6 (cite)

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

$K^\bullet$ represents a compact object of $D(\mathcal{O})$ , and
$E = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O})}(K^\bullet , K^\bullet )$ 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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