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:http://arxiv.org/abs/0706.2420v3 (published in Duke Math. J. 145, 2008)http://arxiv.org/abs/0903.3717v2 (to appear in Compositio Math.)http://arxiv.org/abs/1001.0228v2 (to appear in J. K-theory)as well as quite a few other papers by Tabuada available on the arXiv.