$K$-theory of cones of smooth varieties

TL;DR

This paper computes the algebraic K-theory of the homogeneous coordinate ring of smooth projective varieties, linking it to geometric properties of the embedding, with explicit results for curves.

Contribution

It provides explicit formulas for the K-theory of coordinate rings of smooth varieties, especially curves, connecting algebraic K-theory to geometric cohomology.

Findings

Calculated K_0 and K_1 for curves

Established K_{-1} in terms of H^1 cohomology

Expressed K_0 using Zariski cohomology of twisted differentials

Abstract

Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $\k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=\oplus H^1(C,\cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted K\"ahler differentials on the variety.