Reconstruction algebras of type 𝐴

Wemyss, Michael

doi:10.1090/s0002-9947-2011-05130-5

Cited by 28 publications

(41 citation statements)

References 27 publications

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“…Moreover, moduli spaces of quiver representations provide a very useful technique to obtain commutative resolutions from non-commutative resolutions, see e.g. [91,94].…”

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confidence: 99%

Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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Abstract. Let R be an isolated Gorenstein singularity with a non-commutative resolution A = End R (R ⊕ M ). In this paper, we show that the relative singularity category ∆ R (A) of A has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category D sg (R) of Buchweitz and Orlov as a certain canonical quotient category. If R has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that D sg (R) determines ∆ R (Aus(R)), where Aus(R) is the corresponding Auslander algebra. The proofs of these results use dg algebras, A ∞ Koszul duality, and the new concept of dg Auslander algebras, which may be of independent interest.

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“…Moreover, moduli spaces of quiver representations provide a very useful technique to obtain commutative resolutions from non-commutative resolutions, see e.g. [91,94].…”

mentioning

confidence: 99%

Relative singularity categories I: Auslander resolutions

Kalck

Yang

2016

Advances in Mathematics

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“…It is an improvement of [8,Theorem 3.8], the latter having the additional requirement that R is Auslander-regular. Theorem 3.13 is illustrated by an application to the Reconstruction Algebras of [39] in Section 3.5.…”

Section: 2mentioning

confidence: 99%

The Cohen Macaulay Property for Noncommutative Rings

Brown

Macleod

2017

Algebr Represent Theor

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Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.

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“…For some special finite subgroups G ⊆ GL(2, C), Iyama and Wemyss introduced the reconstruction algebra and generalized Brieskorn's results (see [17], [24], [23], [22], [21]). Thus we can view reconstruction algebras as a natural geometric generalization of preprojective algebras of extended Dynkin diagrams.…”

Section: Introductionmentioning

confidence: 99%

“…The reconstruction algebra plays important roles in algebraic geometry and commutative algebra. In the papers [24], [23], [22], [21], it is shown that the moduli space of finite dimensional representations of a reconstruction algebra in types A and D corresponding to G ⊆ GL(2, C) contains enough information to construct the This work was financially supported by National Natural Science Foundation of China (Grant Nos. 11301143 and 11301144).…”

Section: Introductionmentioning

confidence: 99%

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type A 1

Hou

Guo

2015

Czech Math J

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Abstract. The reconstruction algebra is a generalization of the preprojective algebra, and plays important roles in algebraic geometry and commutative algebra. We consider the homological property of this class of algebras by calculating the Hochschild homology and Hochschild cohomology. Let Λ t be the Yoneda algebra of a reconstruction algebra of type A 1 over a field k. In this paper, a minimal projective bimodule resolution of Λ t is constructed, and the k-dimensions of all Hochschild homology and cohomology groups of Λ t are calculated explicitly.

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Reconstruction algebras of type 𝐴

Abstract: Abstract. We introduce a new class of algebras, called reconstruction algebras, and present some of their basic properties. These non-commutative rings dictate in every way the process of resolving the Cohen-Macaulay singularities 2 /G where G = 1 r (1, a) ≤ GL(2, ).

Cited by 28 publications

References 27 publications

Relative singularity categories I: Auslander resolutions

Relative singularity categories I: Auslander resolutions

The Cohen Macaulay Property for Noncommutative Rings

Hochschild (co)homology of Yoneda algebras of reconstruction algebras of type A 1

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