Lemma 22.33.5 (09LT)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.33: Derived tensor product
Lemma 22.33.5 (cite)

numberstags

Lemma 22.33.5. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded $R$-algebras. Let $N$ be a differential graded $(A, B)$-bimodule. Then the functor

\[ - \otimes _ A^\mathbf {L} N : D(A, \text{d}) \longrightarrow D(B, \text{d}) \]

of Lemma 22.33.2 is a left adjoint to the functor

\[ R\mathop{\mathrm{Hom}}\nolimits (N, -) : D(B, \text{d}) \longrightarrow D(A, \text{d}) \]

of Lemma 22.31.2.

Proof. This follows from Derived Categories, Lemma 13.30.1 and the fact that $- \otimes _ A N$ and $\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, -)$ are adjoint by Lemma 22.30.3. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment