The universal Chern character and a non-commutative Hirzebruch-Riemann-Roch theorem

Denis-Charles Cisinski – 1 June 2011

This is joint work with G. Tabuada.

There is a universal functor from the homotopy category of dg(=differential graded) categories to a stable homotopy theory with the following properties: it inverts derived Morita equivalences, it preserves filtrered homotopy colimits, and sends Drinfeld quotients (aka the dg analogs of Verdier quotients of triangulated categories) to homotopy cofibers. The universal Chern character theorem states that the (derived) Hom's in this target category computes non-connective K-theory (i.e., in positive degrees, we get back Quillen-Waldhausen K-theory, while, in negative degrees, we obtain Bass negative K-groups). Moreover, this universal functor is symmetric monoidal, from which we get a Hirzebruch-Riemann-Roch formula for smooth and proper dg categories.

Main References: (published in Duke Math. J. 145, 2008) (to appear in Compositio Math.) (to appear in J. K-theory)

as well as quite a few other papers by Tabuada available on the arXiv.