Isolated Singularities, Smooth Orders, and Auslander Regularity

Abstract

In this note we prove that a smooth order satisfying the reverse geometric engineering conditions in stringtheory [5, §2] (in any compactifying dimension) is Auslander regular. Moreover, we classify theétale local structure of smooth orders over an isolated central singularity.

Graded Calabi Yau algebras of dimension 3

“…In the cases where (k, d) equals (2,4) or (3,3), one can obtain a complete classification of the good superpotentials, because then we are in the case of Artin-Shelter regular algebras [1].…”

“…On this space, we can define for every arrow a a 'derivation' ∂ a that cuts out a (for a precise definition see Section 2.1). Given a superpotential W , one can construct the vacualgebra [3] A W := CQ/(∂ a W : a ∈ Q 1 ).…”

The singularities of noncommutative manifolds

“…However, we will not draw this arrow so that these types become sinks in the map. This happens when the corresponding quotient variety is an isolated singularity and those quiver-quotients have been classified in [5] to be of the form…”

Noncommutative Resolutions of ADE Fibered Calabi-Yau Threefolds

“…Various steps in the direction mentioned above have been taken in a series of papers [31,30,21,37,42], where several concrete examples have been discussed. More abstract approaches have also been put forward in [29,6]. The lesson to be drawn from these works is that for some singularities it is possible to find a noncommutative algebra A such that the representation theory of this algebra dictates in every way the process of resolving these singularities.…”

“…More abstract approaches have also been put forward in [29,6]. The lesson to be drawn from these works is that for some singularities it is possible to find a noncommutative algebra A such that the representation theory of this algebra dictates in every way the process of resolving these singularities.…”