Preresolutions of noncommutative isolated singularities

Ji-Wei He and Yu Ye

Vol. 316 (2022), No. 2, 367–394

DOI: 10.2140/pjm.2022.316.367

Abstract

We introduce the notion of right preresolutions (quasiresolutions) for noncommutative isolated singularities, which is a weaker version of quasiresolutions introduced by Qin, Wang and Zhang ( J. Algebra 536 (2019), 102–148). We prove that right quasiresolutions for a noetherian bounded below and locally finite graded algebra with right injective dimension 2 are always Morita equivalent. When we restrict to a noncommutative quadric hypersurface $A$, we prove that if $A$ is a noncommutative isolated singularity, then it always admits a right preresolution. 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 preresolutions is included as well.

Keywords

right preresolution, noncommutative isolated singularity, noncommutative quadric hypersurface

Mathematical Subject Classification

Primary: 16E65, 16G50, 16S37

Milestones

Received: 7 September 2020

Revised: 19 July 2021

Accepted: 21 December 2021

Published: 6 April 2022