The space of Penrose tilings and the noncommutative curve with homogeneous coordinate ring \(k\langle x,y \rangle/(y^2)\)
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 sheaves at points of \(\text{Proj}B\). In this text, the chosen objects are \(B\)-modules \(M\) that are simple as an object in \(\mathsf{QGr}B\). A Penrose sequence is a sequence \(z=z_0z_1\dots\) of \(0\)'s and \(1\)'s having no consecutive \(1\)'s. The set of such sequences are denoted \(\mathbf{P}\). There is a one-to-one correspondence between \(\mathbf{P}\) and the set of Penrose tilings, \(z\mapsto\mathcal T_z\), and \(z\sim z^\prime\) if there exists an \(m\) such that \(z_n=z_n^\prime\) for all \(n\geq m\). To a Penrose sequence \(z\), the point module \(M_z\) is defined as \(M_z=ke_0\oplus ke_1\oplus\cdots\), \(\deg e_i=i\), \(xe_i=(1-z_i)e_{i+1}\) and \(ye_i=z_ie_{i+1}\). Then \(y^2e_i=z_iz_{i+1}e_{i+2}=0\), and \(M_z\) is a graded \(B\)-module denoted by \(\mathcal O_z\) when mapped to \(\mathsf{QGr}B\). The main result in this setting is that if \(z\) is a Penrose sequence, then \(\mathcal O_z\) is a simple object in \(\mathsf{QGr}B\) and the following are equivalent: (1) \(\mathcal O_z\simeq\mathcal O_{z^\prime}\), (2) \(z\sim z^\prime\), (3) there is an isometry \(\sigma:\mathbb R^2\rightarrow\mathbb R^2\) such that \(\sigma(\mathcal T_z)=\mathcal T_{z^\prime}\). The author defines degree-and Serre Twist, and gives the algebraic and topological properties of \(\mathsf{QGr}B\). This gives a frame for proving several results to support that \(\text{Proj}_{nc}B\) is a commutative algebraic geometric version of the space of Penrose tilings. This includes dimension, density, and closed subspaces. The author proves that \(\text{Proj}_{nc}B\) is a smooth curve, even if \(B\) is most unlike the rings that appear in classical algebraic geometry. The last topic of the article is the computation of the Grothendieck group. \(K_0(\mathsf{qgr}(B))\) is computed as an ordered abelian group. This is done by working with path algebras, which is a nice illustration of how this can be done. The correspondence with, or rather the use of, the Bratteli diagrams is of course very interesting. The article gives a nice treatment of the algebraic geometry of noncommutative graded rings, and its correspondence with Connes' work and operator algebras. The results are important, illustrative and nice, and give a very good understanding of the theory, working in a implicitly defined noncommutative projective space. Also, an appendix on tilings of the plane is given at the end of the article.
scientific article; zbMATH DE number 6347618
scientific article