The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring \(k\langle x,y \rangle/(y^2)\)
type: Item
instance of: scholarly article
title: The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring \(k\langle x,y \rangle/(y^2)\)
zbMATH Open document ID: 1328.14007
DOI: 10.4171/JNCG/164
published in: Journal of Noncommutative Geometry
publication date: 2014-09-25T00:00:00Z
review text: Let \(B=k\langle x,y\rangle/(y^{r+1})\). The author considers only the case with \(r=1\) in this article, and claims that all results can be extended to \(r>2\). The case \(r=1\) relates to the Penrose tilings. To the space of Penrose tilings, Connes associates the \(C^\ast\)-algebra \(\overline{S}\) which is the limit of some given matrix algebras with order \(f_n\) the Fibonacci numbers, and explicitly given limit morphisms \(\phi_n:S_n\rightarrow S_{n+1}\). The noncommutative projective space \(\text{Proj}_{nc} B\) is defined implicitly by declaring that \(\mathsf{Qcoh}(\text{Proj}_{nc} B)=\mathsf{QGr}B\) with \(\mathsf{QGr}B=\frac{\mathsf{Gr}B}{\mathsf{Fdim}B}\), the category \(\mathsf{Gr}B\) of \(\mathbb Z\)-graded left \(B\)-modules modulo the Serre subcategory of modules that are unions of their finite dimensional submodules. The notation is the quotient functor \(\beta^\ast:\mathsf{Gr}B\rightarrow\mathsf{QGr}B,\) and \(\mathcal O=\beta^\ast B\). Also \(\mathsf{qgr}B=\frac{\mathsf{gr}B}{\mathsf{fdim}B}\), and \(\mathsf{qgr}B\) is equivalent to the full subcategory of \(\mathsf{QGr}B\) consisting of finitely presented objects. That is, \(\mathsf{qgr}B\) is the category of coherent sheaves on \(\text{Proj}_{nc}B\). The main result of the article is that if \(S\) is the direct limit in the category of \(\mathbb C\)-algebras of the directed system \((S_n,\phi_n)\), if \(\mathsf{Mod}S\) is the category of right \(S\)-modules, and \(\mathsf{mod}S\) the full subcategory of finitely presented right \(S\)-modules, then there is an equivalence of categories \(\mathsf{QGr}(\frac{\mathbb C\langle x,y\rangle}{(y^2)})\equiv\mathsf{Mod}S\) that restricts to an equivalence on \(\mathsf{qgr}(\mathbb C\langle x,y\rangle/(y^2)\). Under this equivalence, \(\mathcal O\) corresponds to \(S\). This implies that \(\mathsf{Mod}S\) is a locally coherent category, and \(\mathsf{mod}S\) is an abelian category. Some chosen simple objects in \(\mathsf{QGr}B\) are given as skyscraper sheav…
reviewed by: Arvid Siqveland
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 19K14
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 58B34
zbMATH DE Number: 6347618
zbMATH Keywords: Penrose tilings
zbMATH Keywords: skyscraper sheaves
zbMATH Keywords: AF-algebra
zbMATH Keywords: Grothendieck group
zbMATH Keywords: Penrose sequence
zbMATH Keywords: connected algebra
zbMATH Keywords: Bratteli diagram
author: S. Paul Smith
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6347618