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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

flag	asked Feb 21 2011 at 19:46 Shizhuo Zhang 3,489●1●7●37

Hopefully, someone will give you a detailed answer. Such theorems have certainly been proved by Gillet, Soulé and others. The article by Tamme in the Beilinson conjectures volume may a good place to start. – Donu Arapura Feb 21 2011 at 22:15

profilesdroxford54 · Accepted Answer · 2011-02-22 00:14:44Z UTC

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MR0624666 (83m:14013) Gillet, Henri Riemann-Roch theorems for higher algebraic K-theory. Adv. in Math. 40 (1981), no. 3, 203–289.

answered Feb 22 2011 at 0:14

profilesdroxford54
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	Thanks for the reference – Shizhuo Zhang Feb 22 2011 at 0:31
	There is also the much shorter announcement, also by Gillet, available freely at projecteuclid.org/… – David Roberts Feb 22 2011 at 2:48

David Roberts · Answer 2 · 2011-02-22 04:50:48Z UTC

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.

Grothendieck Riemann Roch involving Higher K ?

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