On Grothendieck's Riemann-Roch Theorem

TL;DR

This paper demonstrates that the $K$-theory of vector bundles on smooth quasi-projective varieties is a universal cohomology theory, and derives Grothendieck's Riemann-Roch theorem from this universal property.

Contribution

It establishes the universality of $K$-theory as a cohomology theory and derives Grothendieck's Riemann-Roch theorem from this property.

Findings

$K$-theory is the universal cohomology theory for vector bundles.

Grothendieck's Riemann-Roch theorem follows from the universal property.

The graded $K$-theory $GK^ullet (X) ensor Q$ also exhibits this universality.

Abstract

We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\otimes \bar L)=c_1(L)+c_1(\bar L)-c_1(L)c_1(\bar L)$. Then, we show that Grothendieck's Riemann-Roch theorem is a direct consequence of this universal property, as well as the universal property of the graded $K$-theory $GK^\bullet (X)\otimes \mathbb{Q}$.