#### Vol. 316, No. 2, 2022

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Preresolutions of noncommutative isolated singularities

### Ji-Wei He and Yu Ye

Vol. 316 (2022), No. 2, 367–394
##### 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
##### Authors
 Ji-Wei He Hangzhou Normal University Hangzhou China Yu Ye University of Science and Technology of China Hefei China