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Noncommutative geometry. A functorial approach. (English) Zbl 1388.14001

De Gruyter Studies in Mathematics 66. Berlin: De Gruyter (ISBN 978-3-11-054317-9/hbk; 978-3-11-054525-8/ebook). xiii, 263 p. (2017).
This book appears at the time Robert Langland is given the Abel prize. It covers much more, but the Langlands program is included.
The book consists of three parts, the first preparatory, and is called Basics. It starts by explaining model examples, that is the simplest functors arising in algebraic geometry, number theory and topology. The functors take values in \(C^\ast\)-algebras, and are the noncommutative tori. The noncommutative torus \(\mathcal A_\theta\) is an algebra over \(\mathbb C\) on a pair of generators \(u,v\) satisfying \(vu=e^{2\pi i\theta}\) where \(\theta\) is a real constant. There are several more or less equivalent definitions of noncommutative tori. The original is the geometric, which involves a deformation of the commutative algebra \(C^\infty(T^2)\) of smooth complex valued functions on a two-dimensional torus \(T^2\). After this comes the analytic definition, and finally the algebraic definition. The different definitions give a good illustration of the connections, and the reason for the more abstract algebraic definition of a torus. Some basic facts about the torus \(\mathcal A_\theta\) are considered. The author gives the identity for when \(\mathcal A_\theta\) is Morita equivalent to \(\mathcal A_{\theta^\prime}\) in the geometric and analytic case, which says when the to tori are equivalent in the setting of noncommutative algebraic geometry. Also, complex and real multiplication, the relation to elliptic curves with Weiserstrass uniformation and Jacobi normal forms are considered. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\tau\) is explained, and this makes the definition of the Sklyanin algebra \(S(\alpha,\beta,\gamma)\) natural. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\theta\) is intertwined with the arithmetic of elliptic curves, and the ranks of elliptic curves are related to an invariant of algebras \(\mathcal A_{RM}\). The first chapter ends with a classification of surface automorphisms.
Via the basics of category theory, the book contains a thorough definition of \(C^\ast\)-algebras: A \(C^\ast\)-algebra is an algebra \(A\) over \(\mathbb C\) with a norm \(a\mapsto\parallel a\parallel\) and an involution satisfying \(\parallel ab\parallel\leq \parallel a\parallel\parallel b\parallel\) and \(\parallel a^\ast a\parallel=\parallel a\parallel^2\) and such that \(A\) is complete with respect to the norm. The properties of \(C^\ast\)-algebras are recalled, and the \(K\)-theory of \(C^\ast\)-algebras is given. After this it is possible to define the noncommutative tori and the almost finite (AF) algebras introduced by Bratteli, which are classified by their Bratteli diagrams. Particular cases of these are generic AF algebras, stationary AF-algebras, and the original case: Uniformly hyper-finite \(C^\ast\)-algebras (UHF-algebras). Also, the \(K\)-groups of the Cuntz-Krieger algebras are computed.
The second part of the book studies noncommutative invariants, first in the case of topology. This is done by constructing functors arising in the topology of surface automorphisms, fibre bundles, knots, links, etc. The functors have their images in the category of AF-algebras, Cuntz-Krieger algebras, cluster \(C^\ast\)-algebras and so forth, defining a set of homotopy invariants of the the corresponding topological space. Some invariants are new, and some are known: Torsion in fibre bundles, Jones and HOMFLY polyniomials, and more. For the classification of surface automorphisms of compact oriented surfaces of genus \(g\geq 1\), the text considers Pseudo-Anasov automorphisms of a surface, the Jacobians of measured foliations, Anosov maps of the torus and its numerical invariants, etc.
In the study of torsion in the torus bundles \(M_\alpha\), the study includes the Cuntz-Krieger functor and specific noncommutative invariants of torus bundles.
In the study of the obstruction theory for Anosov’s bundles, a functor \(F\) from the category of mapping tori of the Anosov diffeomorphisms \(\phi:M\rightarrow M\) of a smooth manifold \(M\), the Anosov bundles, to a category of stable homomorphisms between corresponding AF-algebras is constructed. This is used to introduce an obstruction theory for continuous maps between Anosov’s bundles built on the noncommutative invariants derived from the Handleman triple \((\Lambda,[I],K)\) attached to a stationary AF-algebra. Specific examples are given in dimension 2, 3 and 4.
Now follows the definition, properties and applications of cluster \(C^\ast\)-algebras and knot polynomials in the topological setting. The author gives a representation of the braid groups in the cluster \(C^\ast\)-algebra associated to a triangulation of the Riemann surface \(S\) with one or two cusps, and it is proved that the Laurent polynomials coming from the \(K\)-theory of such an algebra are topological invariants of the closure of braids. Jones and HOMFLY polynomials are special cases of the construction corresponding to the \(S\) being a sphere with two cusps and a torus with one cusp respectively. One should mention that the text covers the Birman-Hilden theorem and a lot of explicit examples.
The book turns over to noncommutative invariants in algebraic geometry: The setup is to look at \(\mathsf{CRng}\) as the category of coordinate rings of projective varieties, and to consider a functor \(F:\mathsf{CRng}\overset{\text{GL}_n} {\rightarrow}\mathsf{Grp}\hookrightarrow\mathsf{Grp-Rng}\). It is proved in the text that when \(\mathsf{CRng}\) are the coordinate rings of elliptic curves, the category \(\mathsf{Grp-Rng}\) are the noncommutative tori. Also, if \(\mathsf{CRng}\) are the rings of algebraic curves of genus \(g\geq 1\), then \(\mathsf{Grp-Rng}\) are the toric AF-algebras. If \(\mathsf{CRng}\) are coordinate rings of projective varieties of dimension \(n\geq 1\), then \(\mathsf{Grp-Rng}\) consists of the Serre \(C^\ast\)-algebras. Notice also that elliptic curves over the field of \(p\)-adic numbers are considered, and in this case \(\mathsf{Grp-Rng}\) consists of the UHF-algebras. Finally in this chapter on noncommutative invariants in algebraic geometry, it is proved that the mapping class group of genus \(g\geq 2\) are linear: They admit a faithful representation into the matrix group \(\text{GL}_{6g-6}(\mathbb Z)\). This chapter includes Elliptic curves, algebraic curves of genus \(g\geq 1\), Tate curves and UHF-algebras and the mapping class group. It should be mentioned that the link between topology and algebraic geometry is explored.
In number theory, noncommutative invariants appear via a restriction of a functor \(F:\mathsf{CRng}\rightarrow\mathsf{Grp-Rng}\) to the arithmetic schemes. An important example is when \(\mathcal E_{\text{CM}}\) is an elliptic curve with complex multiplication. Then \(F(\mathcal E_{\text{CM}})=\mathcal A_{\text{RM}}\) where \(\mathcal A_{\text{RM}}\) is a noncomutative torus with real multiplication. This is used to relate the rank of \(\mathcal E_{\text{CM}}\) to an invariant of \(\mathcal A_{\text{RM}}\). This invariant is called arithmetic complexity, and is used to prove that the complex number \(e^{2\pi i\theta+\log\log\varepsilon}\) is algebraic whenever \(\theta\) and \(\varepsilon\) are algebraic numbers in a real quadratic field. Also, the invariant is used to find generators of the abelian extension of a real quadratic number field. The text includes the definition of an \(L\)-function \(L(\mathcal A_{\text{RM}},s)\) of \(\mathcal A_{\text{RM}}\) and proves that this coincides with the Hasse-Weil function \(L(\mathcal E_{\text{CM}},s)\) of \(\mathcal E_{\text{CM}}\). This localization functor tells that the crossed products is an analogue of the prime ideals used in algebraic geometry. The function \(L(\mathcal A_{\text{RM}},s)\) is extended to the even-dimensional noncommutative tori \(\mathcal A_{\text{RM}}^{2n}\), and an analogue of the Langlands conjecture for such tori is sketched. The number of points of a projective variety \(V(\mathbb F_q)\) over a finite field \(\mathbb F_q\) is computed in terms of the invariants of the Serre \(C^\ast\)-algebra \(F(V_{\mathbb C})\) of the complex projective variety \(V_{\mathbb C}\). Also, this chapter consider isogenies of elliptic curves, symmetry of complex and real multiplication, ranks of elliptic curves, transcendental number theory, class field theory, noncommutative reciprocity, Langlands conjecture for the \(\mathcal A_{\text{RM}}^{2n}\), and finally, projective varieties over finite fields.
The final part of the book gives a survey of Noncommutative algebaic geometry (NCG). The start is the finite geometries and the axioms of projective geometry, including Desargues and Pappus axioms. Then continuous geometries in the weak geometry, von Neumann geometry. In particular, Connes’ geometries is considered more deeply, including classification of type III factors, Connes’ invariants, noncommutative differential geometry and Connes’ index theorem. Chapter 10 contains a survey of Index theory: The Atiyah-Singer theorem and Fredholm operators, the index theorem, \(K\)-homology and Atiyah’s realization, Kasparov’s \(KK\)-theory. The applications of the index theory include the Novikov conjecture, Baum-Connes conhecture, positive scalar curvature and finally the coarse geometry. After this chapter follows a treatment of Jones polynomials, Braids and the trace invariant. The next to final chapter in the book is about quantum groups,and this includes examples of Hopf algebras already in the introduction to this chapter. Also Manin’s quantum plane, Hopf algebras in general, operator algebras and quantum groups are treated. The final chapter of the book is a survey of noncommutative algebraic geometry. This chapter includes the most common theories of Artin, Van den Bergh (turning slightly into derived schemes). It includes the Serre isomorphism, twisted homogeneous coordinate rings, the Sklyanin algebras, and it even mentions the noncommutative algebraic geometry of O.A. Laudal.
Ok, so there is one more chapter, 14, indicating some trends in NCG, but this is most likely separated into its own chapter just because of tridecrafobia.
The book is a good survey of noncommutative geometry, and is an excellent starting point for doing good research in the field. Each section of the book ends with a list of references for going deeply into each subject, and so this book gives a framework for the field of noncommutative geometry.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
18F99 Categories in geometry and topology
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