Regular $\mathbb{Z}$-graded local rings and Graded Isolated Singularities

TL;DR

This paper characterizes regular $Z$-graded local rings and establishes a connection between graded isolated singularities and the global dimension of associated noncommutative projective schemes, linking algebraic and geometric properties.

Contribution

It provides new characterizations of regular $Z$-graded local rings and relates graded isolated singularities to the smoothness of noncommutative projective schemes.

Findings

Regular $Z$-graded local rings are characterized similarly to classical local rings.

Graded isolated singularities correspond to smoothness of associated noncommutative projective schemes.

A graded algebra is a singularity if and only if its projective scheme has finite global dimension.

Abstract

In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated singularity for a commutative $\mathbb{Z}$-graded semilocal algebra in terms of the global dimension of its associated noncommutative projective scheme. As a corollary, we obtain that a commutative affine $\mathbb{N}$-graded algebra generated in degree $1$ is a graded isolated singularity if and only if its associated noncommutative projective scheme is smooth; if and only if the category of coherent sheaves on its projective scheme has finite global dimension, which are known in literature.