K_0 of hypersurfaces defined by x_1^2+ ... + x_n^2 = \pm 1

TL;DR

This paper computes the Grothendieck group K_0 of coordinate rings of hypersurfaces defined by quadratic forms resembling spheres over fields with characteristic not 2, providing explicit algebraic invariants.

Contribution

It explicitly determines the K_0 groups of rings associated with quadratic hypersurfaces defined by x_1^2 + ... + x_n^2 = ±1 over fields of characteristic not 2.

Findings

Calculated K_0 groups for all n,m ≥ 0

Provided algebraic invariants for quadratic hypersurfaces

Extended understanding of algebraic K-theory in this context

Abstract

Let $k$ be a field of characteristic $\ne 2$ and let $Q_{n,m}(x_1, ...,x_n,y_1, ...,y_m)=x_1^2+ ... +x_n^2-(y_1^2+ ... +y_m^2)$ be a quadratic form over $k$. Let $R(Q_{n,m})=R_{n,m}=k[x_1, ...,x_n,y_1, ...,y_m]/(Q_{n,m}-1)$. In this note we will calculate $\wt K_0(R_{n,m})$ for every $n,m \geq 0$.