Non-commutative resolutions of toric varieties
type: Item
instance of: scholarly article
title: Non-commutative resolutions of toric varieties
zbMATH Open document ID: 1423.13075
author: Eleonore Faber
author: Greg Muller
author: K. E. Smith
published in: Advances in Mathematics
publication date: 2019-07-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1805.00492
full work available at URL: 1805.00492.pdf
review text: Let \(R\) be a local or graded, commutative, Noetherian ring. Auslander-Buchsbaum and Serre have proved that \(R\) is regular if and only if \(R\) has finite global dimension. \textit{J. Dixmier} [Anais Acad. Brasil. Ci. 35, 491--519 (1963; Zbl 0143.05302)] made finite global dimension a part of the definition of regularity for non-commutative rings, and the understanding of the connection between regularity and finite global dimension has been refined by Auslander, Artin, Shelter, Van den Bergh and others. In particular, if \(R\) is a regular ring of finite type over a perfect field \(k\), then the ring of differential operators \(D_k(R)\) has finite global dimension. The proof of this result was given by \textit{J.-E. Roos} [C. R. Acad. Sci., Paris, Sér. A 274, 23--26 (1972; Zbl 0227.16021)] in characteristic zero, and \textit{S. P. Smith} [J. Algebra 107, 98--105 (1987; Zbl 0617.13007)] in characteristic \(p\). This paper establishes the finite global dimension of several non-commutative algebras associated to a normal toric algebra \(R\) over a field \(k\). Such \(R\) is commutative, but only rarely regular. The algebras under consideration in this article are (1) \(\operatorname{End}_R(\mathbb{A})\), where \(\mathbb{A}\) is a complete sum of conic modules. Note that a conic module is a combinatorial defined \(R\)-module given below, and a direct sum of conic modules is complete if every conic module is isomorphic to a summand. (2) \(\operatorname{End}_R(R^{1/q})\), for sufficiently large \(q\). \(R^{1/q}\) is the \(k\)-algebra spanned by \(q\)-th roots of monomials in \(R\). When \(k\) is perfect and \(q\) is a power of the characteristic of \(k\), \(R^{1/q}\) is the ring of \(q\)-th roots of elements in \(R\). (3) The ring of \(k\)-linear differential operators \(D_k(R)\) of \(R\), assuming \(k\) is perfect with positive characteristic. The main result of the paper is an explicit projective resolution for each simple module for the algebra \(\operatorname{End}…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 14A22
zbMATH DE Number: 7078808
zbMATH Keywords: regular ring
zbMATH Keywords: global dimension
zbMATH Keywords: regularity for non-commutative rings
zbMATH Keywords: ring of differential operators
zbMATH Keywords: normal toric algebra
zbMATH Keywords: conic module
zbMATH Keywords: complete conic module
zbMATH Keywords: projective resolution
zbMATH Keywords: non-commutative resolution
zbMATH Keywords: non-commutative crepant resolution
zbMATH Keywords: simplicial algebra
zbMATH Keywords: chambers of constancy
zbMATH Keywords: hyperplane arrangement
zbMATH Keywords: acyclicity Lemma
zbMATH Keywords: Frobenius map
zbMATH Keywords: Kunz's Theorem
zbMATH Keywords: F-regularity
zbMATH Keywords: minimal model program
zbMATH Keywords: rational singularities
describes a project that uses: PyFrobenius
MaRDI profile type: Publication
OpenAlex ID: W2963223456
arXiv ID: 1805.00492
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Non-commutative resolutions of toric varieties (DOI)
DOI: 10.1016/J.AIM.2019.04.021