Noncommutative localisation in algebraic K-theory II

In [Amnon Neeman, Andrew Ranicki, Noncommutative localisation in algebraic K-theory I, Geom. Topol. 8 (2004) 1385-1425] we proved a localisation theorem in the algebraic K-theory of noncommutative rings. The main purpose of the current article is to express the general theorem of the previous paper in a more user-friendly fashion, in a way more suitable for applications. In the process we compare our result to the existing theorems in the literature, showing how the previous paper improves all the existing results.It should be pointed out that there have been two very interesting recent preprints on related topics. The reader is referred to the beautiful papers of Krause [Henning Krause, Cohomological quotients and smashing localizations, http://wwwmath.upb.de/~hkrause/publications.html. [8]] and Dwyer [William G. Dwyer, Noncommutative localization in homotopy theory, preprint, http://www.nd.edu/~wgd/. [4]]. Krause studies the lifting of chain complexes and the relation with the telescope conjecture, and Dwyer generalises to the homotopy theoretic framework.     

Additivity for derivator K-theory

We prove the additivity theorem for the K-theory of triangulated derivators. This solves one of the conjectures made by Maltsiniotis in [G. Maltsiniotis, La K-théorie d'un dérivateur triangulé, in: Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack, Amnon Neeman (Eds.), Categories in Algebra, Geometry and Physics, Conference and Workshop in honor of Ross Street's 60th Birthday, in: Contemp. Math., vol. 431, Amer. Math. Soc., 2007, pp. 341-368]. We also review some basic definitions and results in the theory of derivators in the sense of Grothendieck.     

Operations in A-theory

A construction for Segal operations for K-theory of categories with cofibrations, weak equivalences and a biexact pairing is given and coherence properties of the operations are studied. The model for K-theory, which is used, allows coherence to be studied by means of (symmetric) monoidal functors. In the case of Waldhausen A-theory it is shown how to recover the operations used in Waldhausen (Lecture Notes in Mathematics, Vol. 967, Springer, Berlin, 1982, pp. 390-409) for the A-theory Kahn-Priddy theorem. The total Segal operation for A-theory, which assembles exterior power operations, is shown to carry a natural infinite loop map structure. The basic input is the un-delooped model for K-theory, which has been developed from a construction by Grayson and Gillet for exact categories in Gunnarsson et al. (J. Pure Appl. Algebra 79 (1992) 255), and Grayson's setup for operations in Grayson (K-theory (1989) 247). The relevant material from these sources is recollected followed by observations on equivariant objects and pairings. Grayson's conditions are then translated to the context of categories with cofibrations and weak equivalences. The power operations are shown to be well behaved w.r.t. suspension and are extended to algebraic K-theory of spaces. Staying close with the philosophy of Waldhausen (1982) Waldhausen's maps are found. The Kahn-Priddy theorem follows from splitting the “free part” off the equivariant theory. The treatment of coherence of the total operation in A-theory involves results from Laplaza (Lecture Notes in Mathematics, Vol. 281, Springer, Berlin, 1972, pp. 29-65) and restriction to spherical objects in the source of the operation.     

Algebraic K-theory of special groups

Following the introduction of an algebraic K-theory of special groups in [Dickmann and Miraglia, Algebra Colloq. 10 (2003) 149-176], generalizing Milnor's mod 2 K-theory for fields, the aim of this paper is to compute the K-theory of Boolean algebras, inductive limits, finite products, extensions, SG-sums and (finitely) filtered Boolean powers of special groups. A parallel theme is the preservation by these constructions of property [SMC], an analog for the K-theory of special groups of the property “multiplication by l(-1) is injective” in Milnor's mod 2 K-theory (see [Milnor, Invent. Math. 9 (1970) 318-344]).     

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex C-algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological K-theory agree on the completed projective tensor product , and that the obstruction for the comparison map to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1)We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of sub-multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485-496].We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. Every Fréchet ideal is sub-harmonic, but not conversely; for example the Schatten ideal Lp is sub-harmonic for all p>0 but is Fréchet only if p?1. We prove a variant of the exact sequence above which essentially says that if A is a C-algebra and J is sub-harmonic, then the obstruction for the periodicity of K_∗(AC_⊗J) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for H-unital algebras announced in [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485-496].The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal K-theory, due to the first author.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

On the Isomorphism Conjecture in algebraic K-theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.     

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

De la K-théorie algébrique vers la K-théorie hermitienne

In this Note, we introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation ${\psi }^2$ in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring $A?A^{\mathrm{op}}$. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Algebraic K-theory of groups wreath product with finite groups

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example, for discrete subgroups of Lie groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], virtually poly-infinite cyclic groups [F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993) 249-297], Artin braid groups [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526], a class of virtually poly-surface groups [S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press] and virtually solvable linear group [F.T. Farrell, P.A. Linnell, K-Theory of solvable groups, Proc. London Math. Soc. (3) 87 (2003) 309-336]. We extend these results in the sense that if G is a group from the above classes then we prove the conjecture for the wreath product G?H for H a finite group. The need for this kind of extension is already evident in [F.T. Farrell, S.K. Roushon, The Whitehead groups of braid groups vanish, Internat. Math. Res. Notices 10 (2000) 515-526; S.K. Roushon, The Farrell-Jones isomorphism conjecture for 3-manifold groups, math.KT/0405211, K-Theory, in press; S.K. Roushon, The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups, math.KT/0408243, K-Theory, in press]. We also prove the conjecture for some other classes of groups.     

Higher Bott-Chern forms and Beilinson's regulator

In this paper, we prove a Gauss-Bonnet theorem for the higher algebraic K-theory of smooth complex algebraic varieties. To each exact n-cube of hermitian vector bundles, we associate a higher Bott-Chen form, generalizing the Bott-Chern forms associated to exact sequences. These forms allow us to define characteristic classes from K-theory to absolute Hodge cohomology. Then we prove that these characteristic classes agree with Beilinson's regulator map. Oblatum 21-III-1997 & 12-VI-1997     

On uniqueness of characteristic classes

We give an axiomatic characterization of maps from algebraic K-theory. The results apply to a large class of maps from algebraic K-theory to any suitable cohomology theory or to algebraic K-theory. In particular, we obtain comparison theorems for the Chern character and Chern classes and for the Adams operations and λ-operations on higher algebraic K-theory. We show that the Adams operations and λ-operations defined by Grayson agree with the ones defined by Gillet and Soulé.     

Maximal exponents of polyhedral cones (I)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rn. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that if K is an n-dimensional polyhedral cone with m extreme rays then for any K-primitive matrix A, γ(A)?(mA−1)(m−1)+1, where mA denotes the degree of the minimal polynomial of A, and the equality holds only if the digraph (E,P(A,K)) associated with A (as a cone-preserving map) is equal to the unique (up to isomorphism) usual digraph associated with an m×m primitive matrix whose exponent attains Wielandt's classical sharp bound. As a consequence, for any n-dimensional polyhedral cone K with m extreme rays, γ(K)?(n−1)(m−1)+1. Our work answers in the affirmative a conjecture posed by Steve Kirkland about an upper bound of γ(K) for a polyhedral cone K with a given number of extreme rays.     

The localization sequence for the algebraic <Emphasis Type="Italic">K</Emphasis>-theory of topological <Emphasis Type="Italic">K</Emphasis>-theory

We verify a conjecture of Rognes by establishing a localization cofiber sequence of spectra \(K(\mathbb{Z})\to K(ku)\to K(KU) \to\Sigma K(\mathbb{Z})\) for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a dévissage theorem identifying the K-theory of the Waldhausen category of finitely generated finite stage Postnikov towers of modules over a connective \(A_\infty\) ring spectrum R with the Quillen K-theory of the abelian category of finitely generated \(\pi_{0}R\)-modules.     

Homotopy theory of associative rings

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a quasi-isomorphism (or weak equivalence) for rings and shows that—similar to spaces—the derived category obtained by inverting the quasi-isomorphisms is naturally left triangulated. Also, homology theories on rings are studied. These must be homotopy invariant in the algebraic sense, meet the Mayer-Vietoris property and plus some minor natural axioms. To any functor X from rings to pointed simplicial sets a homology theory is associated in a natural way. If X=GL and fibrations are the GL-fibrations, one recovers Karoubi-Villamayor's functors KVi, i>0. If X is Quillen's K-theory functor and fibrations are the surjective homomorphisms, one recovers the (non-negative) homotopy K-theory in the sense of Weibel. Technical tools we use are the homotopy information for the category of simplicial functors on rings and the Bousfield localization theory for model categories. The machinery developed in the paper also allows to give another definition for the triangulated category kk constructed by Cortiñas and Thom [G. Cortiñas, A. Thom, Bivariant algebraic K-theory, preprint, math.KT/0603531]. The latter category is an algebraic analog for triangulated structures on operator algebras used in Kasparov's KK-theory.     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

Lower bounds for the minimum eigenvalue of Hadamard product of an M-matrix and its inverse

For the Hadamard product A ° A⁻¹ of an M-matrix A and its inverse A⁻¹, we give new lower bounds for the minimum eigenvalue of A ° A⁻¹. These bounds are strong enough to prove the conjecture of Fiedler and Markham [An inequality for the Hadamard product of an M-matrix and inverse M-matrix, Linear Algebra Appl. 101 (1988) 1-8].