Noncommutative resolutions using syzygies

Dao, Hailong; Iyama, Osamu; Iyengar, Srikanth B.; Takahashi, Ryo; Wemyss, Michael; Yoshino, Yūji

doi:10.1112/blms.12210

Cited by 3 publications

(1 citation statement)

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“…One of them is the notion of a dimension of a subcategory of mod R, as defined in [11,Definition 3.3]. In recent years, the existence of endomorphism rings of finite global dimension, also known as "noncommutative desingularizations" have attracted a lot of attention, see for instance [4,5,6,7,13,16,22,24].…”

Section: Further Consequences Examples and Questionsmentioning

confidence: 99%

Reflexive modules, self-dual modules and Arf rings

Dao

2021

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We prove a tight connection between reflexive modules over a one-dimensional ring R and its birational extensions that are self-dual as R-modules. Consequently, we show that a complete local reduced Arf ring has finitely many indecomposable reflexive modules up to isomorphism, which can be represented precisely by the local rings infinitely near it. We also characterize Arf rings by the property that any reflexive module is self-dual. We give applications on dimension of subcategories and existence of endomorphism rings of small global dimension (non-commutative desingularizations). Our results indicate striking similarities between Arf rings in dimension one and rational singularities in dimension two from representation-theoretic and categorical perspectives.

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Section: Further Consequences Examples and Questionsmentioning

confidence: 99%

Reflexive modules, self-dual modules and Arf rings

Dao

2021

Preprint

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The Frobenius morphism in invariant theory

Raedschelders

Špenko

Bergh

2019

Advances in Mathematics

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Let R be the homogeneous coordinate ring of the Grassmannian G = Gr(2, n) defined over an algebraically closed field k of characteristic p ≥ max{n − 2, 3}. In this paper we give a description of the decomposition of R, considered as graded R p r -module, for r ≥ 2. This is a companion paper to [16], where the case r = 1 was treated, and taken together, our results imply that R has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators D k (R) is simple, that G has global finite F-representation type (GFFRT) and that R provides a noncommutative resolution for R p r .

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