Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:38:29Z http://mathoverflow.net/feeds/question/14861 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14861/is-there-a-refinement-of-the-hochschild-kostant-rosenberg-theorem-for-cohomology Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology? Ian Shipman 2010-02-10T05:46:40Z 2010-07-22T23:30:17Z <p>The HKR theorem for cohomology in characteristic zero says that if $R$ is a regular, commutative $k$ algebra ($char(k) = 0$) then a certain map <code>$\bigwedge^* Der(R) \to CH^*(R,R)$</code> (where <code>$\wedge^* Der(R)$</code> has zero differential) is a quasi-isomorphism of dg vector spaces, that is, it induces an isomorphism of graded vector spaces on cohomology.</p> <p>Can the HKR morphism be extended to an $A_\infty$ morphism? Is there a refinement in this spirit to make up for the fact that it is not, on the nose, a morphism of dg-algebras?</p> http://mathoverflow.net/questions/14861/is-there-a-refinement-of-the-hochschild-kostant-rosenberg-theorem-for-cohomology/14911#14911 Answer by Tony Pantev for Is there a refinement of the Hochschild-Kostant-Rosenberg theorem for cohomology? Tony Pantev 2010-02-10T15:47:58Z 2010-02-11T01:41:37Z <p>Yes there is. It was noted by Kontsevich long time ago that the HKR quasi-isomorphism on cochains can be corrected to give a quasi-isomorphism of dg-algebras and thus induce an $A_\infty$ quasi-isomorphism of minimal models. The correction is very natural - one needs to compose the HKR map it the contraction by the square root of the Todd class, where the latter is understood as a polynomial of the Atiyah class. This story has been studied in great detail in the past few years and has been generalized further to give Tsygan formality which is a quasi-isomorphism of $\infty$-calculi. This was proven by Dolgushev-Tamarkin-Tsygan and also by Calaque-Rossi-van den Bergh.</p> <p>The literature on the subject is huge but you should get a good sense of the results if you look at this <a href="http://arxiv.org/abs/0901.0069" rel="nofollow">survey</a> by Dolgushev-Tamarkin-Tsygan and at this <a href="http://arxiv.org/abs/0904.4890" rel="nofollow">paper</a> of Calaque-Rossi-van den Bergh. There are also many interesting references listed in these papers, for instance the works of Caldararu on the Mukai pairing.</p>