Noncommutative (Crepant) Desingularizations and the Global Spectrum of Commutative Rings

Dao, Hailong; Faber, Eleonore; Ingalls, Colin

doi:10.1007/s10468-014-9510-y

Cited by 19 publications

(26 citation statements)

References 53 publications

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“…Typical examples are polynomial rings over a field, and regular local rings. The following corollary generalizes, and is inspired by, a result of Buchweitz and Pham (unpublished article), who considered the case

N = k

; see also [, Corollary 5.2]. Corollary Let

R

be an equicodimensional regular ring, and

N

a finite length

R

‐module such that

gldim {End}_{R} false(N false)

is finite.…”

mentioning

confidence: 84%

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Noncommutative resolutions using syzygies

Dao

Iyama

Iyengar

et al. 2018

Bull. London Math. Soc.

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N = k

; see also [, Corollary 5.2]. Corollary Let

R

be an equicodimensional regular ring, and

N

a finite length

R

‐module such that

gldim {End}_{R} false(N false)

is finite.…”

mentioning

confidence: 84%

mentioning

confidence: 86%

Noncommutative resolutions using syzygies

Dao

Iyama

Iyengar

et al. 2018

Bull. London Math. Soc.

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“…4.19. NCRs were first defined in [DITV15] over normal rings, we use here the more general definition of [DFI15].…”

Section: Noncommutative Resolutions Of Singularities and The Mckay Comentioning

confidence: 99%

A McKay correspondence for reflection groups

Buchweitz¹,

Faber²,

Ingalls³

2020

Duke Math. J.

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We construct a noncommutative desingularization of the discriminant of a finite reflection group G as a quotient of the skew group ring A = S * G. If G is generated by order two reflections, then this quotient identifies with the endomorphism ring of the reflection arrangement A(G) viewed as a module over the coordinate ring S G /(∆) of the discriminant of G. This yields, in particular, a correspondence between the nontrivial irreducible representations of G to certain maximal Cohen-Macaulay modules over the coordinate ring S G /(∆). These maximal Cohen-Macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement A(G) viewed as a module over S G /(∆). We identify some of the corresponding matrix factorizations, namely the so-called logarithmic co-residues of the discriminant.In order to show that A is an endomorphism ring, we first view A as a CM-module over the (noncommutative) ring T * H and will use the functor i * : Mod(T * H) − → Mod(R/(∆)) , coming from a standard recollement. For this part we will need that G is a true reflection group, that is, generated by reflections of order 2. Then clearly H ∼ = µ 2 . In order to use the 1 Most of our results also hold if the characteristic of the field K does not divide the order |G| of the group G. However, in order to facilitate the presentation, we restrict to K = C.which gives us that the map S J − → S decomposes into components of the form Hom KG (U, S) ⊗ K U J − → Hom KG (U ⊗ det, S) ⊗ K U ⊗ det for each irreducible representation U of G.

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“…In Van den Bergh's original definition, R was assumed to be a commutative normal Gorenstein domain. The reasoning behind this more general definition can be found in [DFI15].…”

Section: (Noncommutative) Resolutions Of Singularitiesmentioning

confidence: 99%