Non-commutative desingularization of determinantal varieties. I
type: Item
instance of: scholarly article
title: Non-commutative desingularization of determinantal varieties. I
zbMATH Open document ID: 1204.14003
author: Ragnar-Olaf Buchweitz
author: Graham J. Leuschke
author: Michel Van den Bergh
published in: Inventiones Mathematicae
publication date: 2010-09-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0911.2659
review text: Let \(K\) be a field and \(X=(x_{ij})\) a \((m\times n)\)-matrix of indeterminates over \(K\), \(n\geq m\). With \(S=K[x_{ij}]\), \(X\) determines the generic \(S\)-linear map \(\phi:S^n\rightarrow S^m.\) Let \(\text{Spec}R\) be the locus in \(\text{Spec}S\) where \(\phi\) has non-maximal rank: \(R\) is the quotient of \(S\) given by the maximal minors of \(X\), and is the generic determinantal variety. The classical \(R\)-modules \(M_a=\text{cok}\bigwedge^a_S\phi\) are maximal Cohen-Macaulay and are resolved by the Buchsbaum-Rim complex. In this article the authors prove that the \((M_a)_a\) yields a kind of non-commutative desingularization of the singular variety \(\text{Spec} R\): For \(1\leq a\leq m\) put \(M_a=\text{cok}\bigwedge^a_S\phi\) and \(M=\bigoplus_a M_a\). Then \(E=\text{End}_R(M)\) is maximal Cohen-Macaulay as an \(R\)-module with finite global dimension. That is, \(E\) is a non-commutative desingularization of \(\text{Spec} R\). If \(m=n\) then \(R\) is the hypersurface \(R=S/(\det\phi)\) and so \(R\) is Gorenstein and the non-commutative desingularization is an example of a non-commutative crepant resolution. The authors give a description by generators and relations of the non-commutative resolution \(E\) by stating that \(E\) as a \(K\)-algebra is isomorphic to the path algebra \(K\tilde Q\) of some quiver \(\tilde Q\). The results above are purely algebraic, but are proved by relating them to algebraic geometry. The classical fact that \(\text{Spec} R\) has a Springer type resolution of singularities is frequently used: Define the incidence variety \[ \mathcal Z=\{([\lambda],\theta)\in\mathbb P^{m-1}(K)\times M_{m\times n}(K)|\lambda\theta=0\} \] with projections \(p^\prime:\mathcal Z\rightarrow\mathbb P^{m-1}\) and \(q^\prime:\mathbb Z\rightarrow\text{Spec} R\). The key geometric facts then include: The scheme \(\mathcal Z\) is projective over \(\text{Spec} R\), which is of finite type over \(K\). The \(\mathcal O_{\mathcal Z}\)-module \[ \mat…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14C40
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 5782773
zbMATH Keywords: non-commutative desingularization
zbMATH Keywords: sparse spectral sequence
zbMATH Keywords: simple representations
zbMATH Keywords: tautological Koszul complex
zbMATH Keywords: maximal Cohen-Macaulay
zbMATH Keywords: quivers
zbMATH Keywords: Clifford algebra
describes a project that uses: PieriMaps
MaRDI profile type: Publication
OpenAlex ID: W1975068388
arXiv ID: 0911.2659
cites work: Q5490943
cites work: HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES
cites work: A Generalized Koszul Complex. II. Depth and Multiplicity
cites work: Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant
cites work: Ideals defined by matrices and a certain complex associated with them
cites work: Resultants and Chow forms via exterior syzygies
cites work: Q5809578
cites work: Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras
cites work: Derived equivalences by quantization
cites work: Q2770573
cites work: Endomorphism Rings of Finite Global Dimension
cites work: Q4215784
cites work: Non-commutative Donaldson-Thomas invariants and the conifold
cites work: Q4669640
cites work: Three-dimensional flops and noncommutative rings
cites work: Generic maps revised
cites work: Q4427516
DOI: 10.1007/S00222-010-0258-7
description: scientific article