On Grothendieck's Riemann-Roch Theorem

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 $G{K}^{\bullet }(X)\otimes \mathbb{Q}$.

18 pages