Lemma 22.37.1 (09S6)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.37: Equivalences of derived categories
Lemma 22.37.1 (cite)

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Lemma 22.37.1. Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded algebras over $R$ , which induces an isomorphism on cohomology algebras. Then

$- \otimes _ A^\mathbf {L} B : D(A, \text{d}) \to D(B, \text{d})$

gives an $R$ -linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \mapsto N_ A$ .

Proof. By Lemma 22.33.7 the functor $M \longmapsto M \otimes _ A^\mathbf {L} B$ is fully faithful. By Lemma 22.33.5 the functor $N \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, N) = N_ A$ is a right adjoint, see Example 22.33.6. It is clear that the kernel of $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2. $\square$

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