$G_0$ of affine, simplicial toric varieties

TL;DR

This paper investigates the structure of the Grothendieck group of coherent sheaves on affine, simplicial toric varieties, revealing its decomposition and the finite nature of certain subgroups, with explicit results in low dimensions.

Contribution

It provides a detailed analysis of the $G_0$ group for affine, simplicial toric varieties, including explicit descriptions and orders of related Chow groups in low dimensions.

Findings

$G_0(X)$ decomposes as $Z igoplus F^1G_0(X)$ with $F^1G_0(X)$ finite.

In dimension 2, $F^1G_0(X)$ is cyclic and its order is determined.

In dimension 3, $F^1G_0(X)$ relates to Chow groups, and the order of $A^1(X)$ is computed.

Abstract

Let $X$ be an affine, simplicial toric variety over a field. Let $G_0$ denote the Grothendieck group of coherent sheaves on a Noetherian scheme and let $F^1G_0$ denote the first step of the filtration on $G_0$ by codimension of support. Then $G_0(X)\cong\mathbb{Z}\oplus F^1G_0(X)$ and $F^1G_0(X)$ is a finite abelian group. In dimension 2, we show that $F^1G_0(X)$ is a finite cyclic group and determine its order. In dimension 3, $F^1G_0(X)$ is determined up to a group extension of the Chow group $A^1(X)$ by the Chow group $A^2(X)$. We determine the order of the Chow group $A^1(X)$ in this case. A conjecture on the orders of $A^1(X)$ and $A^2(X)$ is formulated for all dimensions.