As we know, Grothendieck Riemann Roch only involves $K_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula?
Thanks in advance
As we know, Grothendieck Riemann Roch only involves $K_{0}$. Is there any work generalizing this formula to (Quillen's Higher K)? If there is, what is the meaning for such kind of formula?
Thanks in advance
MR0624666 (83m:14013) Gillet, Henri Riemann-Roch theorems for higher algebraic K-theory. Adv. in Math. 40 (1981), no. 3, 203–289.
There is the recent paper Algebraic K-theory, $A^1$-homotopy and Riemann-Roch theorems by Riou, which gives a different proof of the results by Gillet referenced in the answer by profilesdroxford54.
Abstract:
In this article, we show that the combination of the constructions done in SGA 6 and the $A^1$-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.
To give you a rough idea, algebraic K-theory is representable in the stable homotopy category arising from $A^1$-homotopy theory of schemes over a regular scheme. As far as I understand it, Grothendieck-Riemann-Roch is just a shadow of the maps from the spectrum representing algebraic K-theory to the Eilenberg-MacLane spectrum. But the paper above considers more than just stable homotopy.