Modules over regular algebras of dimension 3
type: Item
instance of: scholarly article
title: Modules over regular algebras of dimension 3
zbMATH Open document ID: 0763.14001
DOI: 10.1007/BF01243916
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143945
review text: A finitely generated graded algebra \(A=k+A_ 1+A_ 2+\cdots\), where \(k\) is a field, is said to be \textit{regular} if \(A\) has finite global dimension and polynomial growth and is Gorenstein. In an earlier work [in The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 33-85 (1990; Zbl 0744.14024)] the authors had classified regular algebras of global dimension 3 which are standard (i.e. generated in degree 1) by establishing a correspondence between such algebras and ``regular'' triples \((E,\sigma,L)\), with \(\sigma\) an automorphism of the scheme \(E\), of one of the following four types: (1a) \(E\) is a cubic divisor in \(\mathbb{P}^ 2\) and \(L={\mathcal O}_ E(1)\), (1b) \(E\) is a divisor of bidegree (2,2) in \(\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L=pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\), (2a) \(E=\mathbb{P}^ 2\) and \(L\cong{\mathcal O}_ E(1)\). (2b) \(E=\mathbb{P}^ 1\times\mathbb{P}^ 1\) and \(L\cong pr^*_ 1({\mathcal O}_{\mathbb{P}^ 1}(1))\). The regularity of the triple means that \(L^{(\sigma-1)(\sigma^ j- 1)}\cong{\mathcal O}_ E\) with \(j=1\) in case (a) and \(j=2\) in case (b). One of the main results proved in the present paper is theorem II: The regular algebra \(A\) is a finite module over its centre if and only if the automorphism \(\sigma\) is of finite order. Given the triple \((E,\sigma,L)\) the corresponding regular algebra \(A\) is obtained via an intermediary graded algebra, namely \(B=\sum_{n\geq 0}H^ 0(E,L\otimes L^ \sigma\otimes\cdots\otimes L^{\sigma^{n- 1}})\). --- The algebras \(A\) and \(B\) are related by \(B\cong A/gA\) with \(g\) homogeneous. Since \(B\) has an explicit description, theorem II is easy to prove for \(B\). In order to deduce the corresponding result for \(A\), the authors introduce the \(\mathbb{Z}\)-graded ring \(\Lambda=A[g^{-1}]\). In analogy with the commutative case, one can think of the non-commutative affine scheme \(\text{Spec} \Lambda_ 0\) as the ``open complement'' of the non-commutative \(\text{P…
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 16E10
zbMATH DE Number: 59805
zbMATH Keywords: regular algebras of global dimension 3
zbMATH Keywords: non-commutative affine scheme
zbMATH Keywords: point modules
zbMATH Keywords: twisting a graded algebra
author: John Tate
author: Michel Van den Bergh
author: Michael Artin
reviewed by: Balwant Singh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59805