Some examples of two-dimensional regular rings

TL;DR

This paper investigates the conditions under which a specific class of two-dimensional rings, defined by quadratic relations over a base ring B, are smooth and regular, with particular focus on rings of algebraic integers.

Contribution

It provides new insights into the smoothness and regularity properties of these quadratic rings, extending understanding in algebraic geometry and number theory.

Findings

Criteria for smoothness of the rings over B

Conditions for regularity when B is a ring of algebraic integers

Examples illustrating the theoretical results

Abstract

Let B be a ring and $A=B[X,Y]/(aX^2+bXY+cY^2-1)$ where $a,b,c\in B$. We study the smoothness of A over B, and the regularity of B when B is a ring of algebraic integers.