Lemma 22.33.4. 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 which has property (P) as a left differential graded $A$-module. Then $M \otimes _ A^\mathbf {L} N$ is computed by $M \otimes _ A N$ for all differential graded $A$-modules $M$.
Proof. Let $f : M \to M'$ be a homomorphism of differential graded $A$-modules which is a quasi-isomorphism. We claim that $f \otimes \text{id} : M \otimes _ A N \to M' \otimes _ A N$ is a quasi-isomorphism. If this is true, then by the construction of the derived tensor product in the proof of Lemma 22.33.2 we obtain the desired result. The construction of the map $f \otimes \text{id}$ only depends on the left differential graded $A$-module structure on $N$. Moreover, we have $M \otimes _ A N = N \otimes _{A^{opp}} M = N \otimes _{A^{opp}}^\mathbf {L} M$ because $N$ has property (P) as a differential graded $A^{opp}$-module. Hence the claim follows from Lemma 22.33.3. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)