Noncommutative resolutions of
                    discriminants

Buchweitz, Ragnar-Olaf; Faber, Eleonore; Ingalls, Colin

doi:10.1090/conm/705/14199

Cited by 8 publications

(7 citation statements)

References 38 publications

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“…This paper evolved from our investigations about a McKay correspondence for reflection groups [BFI17], relating the irreducible representations of a finite complex reflection group to certain maximal Cohen-Macaulay modules over its discriminant. In dimension n = 2 this complements the classical McKay correspondence that connects the irreducible representations of a finite group G SL(2, C) to the cohomology of the minimal resolution of the quotient singularity C 2 /G, see [Buc12,BFI18,GSV83,Rei02] for more details.…”

Section: Introductionmentioning

confidence: 76%

The magic square of reflections and rotations

Buchweitz¹,

Faber²,

Ingalls³

2018

Preprint

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Section: Introductionmentioning

confidence: 76%

The magic square of reflections and rotations

Buchweitz¹,

Faber²,

Ingalls³

2018

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“…The theory of matrix factorisations à la Eisenbud [Eis80, Chapter 6] and Knörrer's method [Knö87, Proposition 2.1] carry over to our setting painlessly. We will use the following re-interpretation, which is also used in [BFI17]. In particular, A has finite representation type if and only if Λ does.…”

Section: Knörrer's Methodsmentioning

confidence: 99%

Low dimensional orders of finite representation type

Chan

Ingalls

2018

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In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders [CI05]. These were classified independently by Artin (in terms of ramification data) and Reiten-Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups G ⊂ GL 2 , explicitly computing H 2 (G, k * ), and then matching these up with Artin's list of ramification data and Reiten-Van den Bergh's AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in [CHI09] to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let B = k ζ x, y be the skew power series ring where ζ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form A = B/(f ) where f ∈ Z(B) which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

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“…Denoting S = Sym C (K n ), this correspondence is realized by the isomorphism of a quotient of the skew group ring S * G with an endomorphism ring over the coordinate ring of the discriminant S G /( ). See also [15] for a more leisurely introduction. This result is of the same flavor as Auslander's algebraic McKay correspondence [3]: he showed that for a small subgroup G ⊆ GL(n, K) the skew group ring S * G is isomorphic to End R (S), where R = S G is the invariant ring.…”

Section: ) Let ([α] δ) and ([β] δ ) Be Two Vertices Of (G) Then There Is An Arrow With Source ([α] δ) And Target ([β] δ ) Whenever [β]mentioning

confidence: 99%

McKay Quivers and Lusztig Algebras of Some Finite Groups

Buchweitz

Faber

Ingalls

et al. 2021

Algebr Represent Theor

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We are interested in the McKay quiver Γ(G) and skew group rings A ∗G, where G is a finite subgroup of GL(V ), where V is a finite dimensional vector space over a field K, and A is a K −G-algebra. These skew group rings appear in Auslander’s version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups G(r,p,n). We first look at the case G(1,1,n), which is isomorphic to the symmetric group Sn, followed by G(r,1,n) for r > 1. Then, using Clifford theory, we can determine the McKay quiver for any G(r,p,n) and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde {A}(\mathsf {G})$ A ~ ( G ) of a finite group $\mathsf {G} \subseteq \text {GL}(V)$ G ⊆ GL ( V ) , which is Morita equivalent to the skew group ring A ∗G. This description gives us an embedding of the basic algebra Morita equivalent to A ∗ G into a matrix algebra over A.

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Noncommutative resolutions of discriminants

Cited by 8 publications

References 38 publications

The magic square of reflections and rotations

The magic square of reflections and rotations

Low dimensional orders of finite representation type

McKay Quivers and Lusztig Algebras of Some Finite Groups

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