Noncommutative Artin motives

In this article, we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category ${{\mathrm{AM}}}(k)_\mathbb Q $ of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives, we then show that the image of this full embedding, which we call the category ${{\mathrm{NAM}}}(k)_\mathbb Q $ of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the symmetry isomorphism constraints. As an application, we recover the absolute Galois group $\mathrm{Gal}(\overline{k}/k)$ from the Tannakian formalism applied to ${{\mathrm{NAM}}}(k)_\mathbb Q $ . Then, we develop the base-change formalism in the world of noncommutative pure motives. As an application, we obtain new tools for the study of motivic decompositions and Schur/Kimura finiteness. Making use of this theory of base-change, we then construct a short exact sequence relating $\mathrm{Gal}(\overline{k}/k)$ with the noncommutative motivic Galois groups of k and $\overline{k}$ . Finally, we describe a precise relationship between this short exact sequence and the one constructed by Deligne–Milne. In the mixed world, we introduce the triangulated category ${{\mathrm{NMAM}}}(k)_\mathbb Q $ of noncommutative mixed Artin motives and construct a faithful functor from the classical category ${{\mathrm{MAM}}}(k)_\mathbb Q $ of mixed Artin motives to it. When k is a finite field, this functor is an equivalence. On the other hand, when k is of characteristic zero ${{\mathrm{NMAM}}}(k)_\mathbb Q $ is much richer than ${{\mathrm{MAM}}}(k)_\mathbb Q $ since its higher Ext-groups encode all the (rationalized) higher algebraic $K$ -theory of finite étale k-schemes. In the appendix, we establish a general result about short exact sequences of Galois groups which is of independent interest. As an application, we obtain a new proof of Deligne–Milne’s short exact sequence.     

KMS weights on higher rank buildings

We extend some of the results of Carey-Marcolli-Rennie on modular index invariants of Mumford curves to the case of higher rank buildings. We discuss notions of KMS weights on buildings, that generalize the construction of graph weights over graph C*-algebras.     

The Ricci Flow on Noncommutative Two-Tori

In this paper we construct a version of Ricci flow for noncommutative two-tori, based on a spectral formulation in terms of the eigenvalues and eigenfunction of the Laplacian and recent results on the Gauss?CBonnet theorem for noncommutative tori.     

Twisted Index Theory on Good Orbifolds, II:¶Fractional Quantum Numbers

This paper uses techniques in noncommutative geometry as developed by Alain Connes [Co2], in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, continuing our earlier work [MM]. We also compute the range of the higher cyclic traces on K-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes–Kubo Hall conductance in the discrete model of the quantum Hall effect on the hyperbolic plane, generalizing earlier results in [Bel+E+S], [CHMM]. The new phenomenon that we observe in our case is that the Connes–Kubo Hall conductance has plateaux at integral multiples of a fractional valued topological invariant, namely the orbifold Euler characteristic. Moreover the set of possible fractions has been determined, and is compared with recently available experimental data. It is plausible that this might shed some light on the mathematical mechanism responsible for fractional quantum numbers. Received: 4 November 1999 / Accepted: 22 September 2000