Pre-resolutions of noncommutative isolated singularities

TL;DR

This paper introduces the concept of right pre-resolutions for noncommutative isolated singularities, establishing conditions for their existence and providing methods for verification, with detailed examples and computations.

Contribution

It defines right pre-resolutions for noncommutative singularities, proves their Morita equivalence under certain conditions, and offers a method to identify isolated singularities in noncommutative quadric hypersurfaces.

Findings

Right quasi-resolutions are Morita equivalent for certain graded algebras.

Noncommutative quadric hypersurfaces with isolated singularities admit right pre-resolutions.

Provided a method to verify isolated singularities in noncommutative quadric hypersurfaces.

Abstract

We introduce the notion of right pre-resolutions (quasi-resolutions) for noncommutative isolated singularities, which is a weaker version of quasi-resolutions introduced by Qin-Wang-Zhang. We prove that right quasi-resolutions for noetherian bounded below and locally finite graded algebra with right injective dimension 2 are always Morita equivalent. When we restrict to noncommutative quadric hypersurfaces, we prove that a noncommutative quadric hypersurface, which is a noncommutative isolated singularity, always admits a right pre-resolution. Besides, we provide a method to verify whether a noncommutative quadric hypersurface is an isolated singularity. An example of noncommutative quadric hypersurfaces with detailed computations of indecomposable maximal Cohen-Macaulay modules and right pre-resolutions is included as well.