4
$\begingroup$

Let $R,S$ be dg-algebras and $f:R \rightarrow S$ be a quasi-isomorphism. Then $f$ induces an equivalence between their derived categories of dg-modules.

If in addition $R,S$ are graded commutative dg-algebras, then their derived categories are equipped with tensor products and the equivalence respects them.

Now suppose one has only a Zig-Zag of quasiisomorphisms, with $S,R$ still graded commutative, but $X$ is not:

$$R \leftarrow X \rightarrow S$$

This induces still an equivalence between the derived cateogies of $R$ and $S$. However will this equivalence be compatible with the monoidal structure?

Put in another way: Let $X$ be a noncommutative dg-algebra, which is known to be quasiisomorphic to a commutative one. Is there a natural monoidal structure on the derived category of $X$?

$\endgroup$
1
  • $\begingroup$ In the context of your last paragraph (and working over a field, for safely reasons): $X$ will be commutative up to coherent homotopies, that is, an $E_\infty$-algebra. If you construct the derived category of $X$ taking into account the higher structure, I am pretty sure tha answer is yes. $\endgroup$ Feb 7, 2012 at 18:52

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.