Lemma 22.34.3. Let $R$ be a ring. Let $(A, \text{d})$, $(B, \text{d})$, and $(C, \text{d})$ be differential graded $R$-algebras. If $C$ is K-flat as a complex of $R$-modules, then (22.34.1.1) is an isomorphism and the conclusion of Lemma 22.34.2 is valid.

**Proof.**
Choose a quasi-isomorphism $P \to (A \otimes _ R B)_ B$ of differential graded $B$-modules, where $P$ has property (P). Then we have to show that

is a quasi-isomorphism. Equivalently we are looking at

This is a quasi-isomorphism if $C$ is K-flat as a complex of $R$-modules by More on Algebra, Lemma 15.59.2. $\square$

## 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Your email address will not be published.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)