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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

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    $\begingroup$ 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. $\endgroup$
    – Donu Arapura
    Commented Feb 21, 2011 at 22:15

2 Answers 2

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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.

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  • $\begingroup$ There is also the much shorter announcement, also by Gillet, available freely at projecteuclid.org/… $\endgroup$
    – David Roberts ♦
    Commented Feb 22, 2011 at 2:48
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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.

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