The Stacks project

Lemma 22.37.1. Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded algebras over $R$, which induces an isomorphism on cohomology algebras. Then

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

gives an $R$-linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \mapsto N_ A$.

Proof. By Lemma 22.33.7 the functor $M \longmapsto M \otimes _ A^\mathbf {L} B$ is fully faithful. By Lemma 22.33.5 the functor $N \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, N) = N_ A$ is a right adjoint, see Example 22.33.6. It is clear that the kernel of $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2. $\square$

Comments (0)

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09S6. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 22.37.1 as pdf