Lemma 24.28.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. If $\mathcal{N}$ is good as a left differential graded $\mathcal{A}$-module, then we have $\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} = \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ for all differential graded $\mathcal{A}$-modules $\mathcal{M}$.

**Proof.**
Let $\mathcal{P} \to \mathcal{M}$ be a quasi-isomorphism where $\mathcal{P}$ is a good (right) differential graded $\mathcal{A}$-module. To prove the lemma we have to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ is a quasi-isomorphism. The cone $C$ on the map $\mathcal{P} \to \mathcal{M}$ is an acyclic right differential graded $\mathcal{A}$-module. Hence $C \otimes _\mathcal {A} \mathcal{N}$ is acyclic as $\mathcal{N}$ is assumed good as a left differential graded $\mathcal{A}$-module. Since $C \otimes _\mathcal {A} \mathcal{N}$ is the cone on the maps $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ as a complex of $\mathcal{O}$-modules we conclude.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)