Lemma 22.31.4 (0CS5)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.31: Derived Hom
Lemma 22.31.4 (cite)

numberstags

Lemma 22.31.4. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over a ring $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then for every $n \in \mathbf{Z}$ there are isomorphisms

\[ H^ n(R\mathop{\mathrm{Hom}}\nolimits (N, M)) = \mathop{\mathrm{Ext}}\nolimits ^ n_{D(B, \text{d})}(N, M) \]

of $R$-modules functorial in $M$. It is also functorial in $N$ with respect to the operation described in Lemma 22.31.3.

Proof. In the proof of Lemma 22.31.2 we have seen

\[ R\mathop{\mathrm{Hom}}\nolimits (N, M) = \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, I) \]

as a differential graded $A$-module where $M \to I$ is a quasi-isomorphism of $M$ into a differential graded $B$-module with property (I). Hence this complex has the correct cohomology modules by Lemma 22.22.3. We omit a discussion of the functorial nature of these identifications. $\square$

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