Lemma 15.59.15 (0BYJ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.59: Derived tensor product
Lemma 15.59.15 (cite)

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Lemma 15.59.15. Let $R$ be a ring. Let $K^\bullet , L^\bullet , M^\bullet$ be complexes of $R$ -modules. There is a canonical isomorphism

$(K^\bullet \otimes _ R^\mathbf {L} L^\bullet ) \otimes _ R^\mathbf {L} M^\bullet = K^\bullet \otimes _ R^\mathbf {L} (L^\bullet \otimes _ R^\mathbf {L} M^\bullet )$

functorial in all three complexes.

Proof. Replace the complexes by K-flat complexes and use the associativity constraint in Section 15.58. $\square$

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