## Abstract

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D-sg(X) and its idempotent completion , we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras (= MMAs), introduced in [20], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to Q-factorial terminalizations in dimension three are MMAs. As an application, we extend some of the algebraic results in [6] and [14] using geometric arguments. (c) 2014 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 85-121 |

Number of pages | 37 |

Journal | Advances in Mathematics |

Volume | 261 |

DOIs | |

Publication status | Published - 20 Aug 2014 |

## Keywords

- Minimal models
- Noncommutative resolutions
- CM modules
- HYPERSURFACE SINGULARITIES
- TRIANGULATED CATEGORIES
- CANONICAL SINGULARITIES
- LOCAL HYPERSURFACES
- TILTING MODULES
- PICARD-GROUPS
- FLOPS
- EQUIVALENCES
- RESOLUTION
- DIMENSION