ContinuousK-theory

We study arithmetical properties of homotopy groups of thel-adic completion of Quillen'sK-theory space of number field, with a view on the Dwyer-Friedlander comparison map into étaleK-theory. The relation of these groups toK-theory is a complete analogy to the relation of continuous étale cohomology to étale cohomology. We identify the torsion subgroup of the resulting term with the subgroup of divisible elements inK _2n (F). We prove that this term is sent isomorphically into étaleK-theory, giving some further evidence for the Lichtenbaum-Quillen conjectures.     

Algebraic K-Theory and the Conjectural Leibniz K-Theory

The analogy between algebraic K-theory and cyclic homology is used to build a program aiming at understanding the algebraic K-theory of fields and the periodicity phenomena in algebraic K-theory. In particular, we conjecture the existence of a Leibniz K-theory which would play the role of Hochschild homology. We propose a motivated presentation for the Leibniz K ₂-group ofa field.     

Generic representations of the finite general linear groups and the Steenrod algebra: II

The category of generic representations over the finite fieldF _q , used in PartI to study modules over the Steenrod algebra, is here related to the modular representation theory of the groups GL _n (F _q ). This leads to a simple and elegant approach to the classic objects of study: irreducible representations, extensions of modules, homology stability, etc. Connections to current research in algebraicK-theory involving stableK-theory and Topological Hochschild Homology are also explained.Partially funded by the NSF.     

On Cyclotomic Numbers and the Reduction Map for the K-Theory of the Integers

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K _2n+1() K_{2n+1}(F_p)is nontrivial. Here n is an even, positive integer and F_p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.     

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then K^G₀(A) @ K^G₀(F){K^G_0(A) \cong K^G_0(F)}, generalizing the analogous result for a dvr by Serre (Inst Hautes études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.     

Semi-Topological K-Homology and Thomason''s Theorem

In this paper, we introduce the 'semi-topological K-homology' of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasi-projective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y ^an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the 'Bott inverted' semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X ^an. In combination with a result of Friedlander and the author, this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that 'Bott inverted' algebraic K-theory with /n coefficients coincides with topological K-theory with /n coefficients.     

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.     

Topological K-Theory of Algebraic K-Theory Spectra

Let KX denote the algebraic K-theory spectrum of a regular Noetherian scheme X. Under mild additional hypotheses on X, we construct a spectral sequence converging to the topological K-theory of KX. The spectral sequence starts from the étale homology of X with coefficients in a certain copresheaf constructed from roots of unity. As examples we consider number rings, number fields, local fields, smooth curves over a finite field, and smooth varieties over the complex numbers.     

KK-groups for generalized operator algebras. II

The extension of Kasparovs bivariant K-theory to inverse limits of C ^* -algebras admits exact Puppe sequences in both variables. Two exact sequences generalizing Milnor's lim-lim¹ sequences are established. For CW complexes the extended K-theory is representable K-theory.     

Dihedral homology and Hermitian K-theory

We establish a rational isomorphism between certain relative versions of Hermitian K-theory and the dihedral homology of simplicial Hermitian rings. This is the dihedral analogue of Goodwillie's result for cyclic homology and algebraic K-theory. In particular, we describe involutions on (negative) cyclic homology and the K-theory of simplicial rings. We show that Goodwillie's map from K-theory to negative cyclic homology can be chosen to preserve involutions. By work of Burghelea and Fiedorowicz the invariants of the involution on K-theory can be identified with symmetric Hermitian K-theory. Finally, we show how the author's chain complex defining dihedral homology can be extended to the left to capture the invariants of the involution on negative cyclic homology.Supported by New Mexico State University, grant No. RC90-051.     

The Algebraic K-Theory of Operator Algebras

We the study the algebraic K-theory of C *-algebras, forgetting the topology. The main results include a proof that commutative C*-algebras are K-regular in all degrees (that is, all theirN ^T K _iand extensions of the Fischer-Prasolov Theorem comparing algebraic and topological K-theory with finite coefficients.     

Morava K-Theories of p-Groups with Cyclic Maximal Subgroups and Other Related Groups

Let P be a non-Abelian finite p-group, p odd, with cyclic maximal subgroups, and let K(n)^*(–) denote the nth Morava K-theory at p. In this paper we determine the algebras K(n)^*(BP) and K(n)^*(BG) for all groups G with Sylow p-subgroups isomorphic to P, giving further evidence for the fact that Morava K-theory as an invariant of finite groups, is finer than ordinary modp cohomology. Mathematics Subject Classifications (2000): 55N20, 55N22.     

Homotopy after tensoring with uniformly hyperfiniteC*-algebras

The paper is devoted to the homotopy classification ofC*-algebras of continuous functions on a finite CW-complex with values in a UHF-algebra. The relevant invariants are based on (connective)K-theory.     

On a spectral sequence for equivariant K-theory

We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin. The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS-0140445 and DMS-0457195.     

On the algebraicK-theory of infinite product categories

Although theK-theory functor on the category of symmetric monoidal categories preserves finite products for essentially trivial reasons, this is not so in the case of infinite products. In this paper, we show that in factK-theory does preserve infinite products, but for non-trivial reasons.Supported in part by NSF DMS 9209714.     

Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

On the algebraic K-theory of the complex K-theory spectrum

Let p≥5 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. In this paper we study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v ₁) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over the polynomial algebra \mathbbF_p[b]{\mathbb{F}}_{p}[b], where b is a class of degree 2p+2 defined as a “higher Bott element”.     

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.     

Negative K-Theory of Generalized Quaternion Groups and Binary Polyhedral Groups

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K _?1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.     

Bivariant K-Theories for C*-Algebras

A new framework for bivariant K-theory is developed. Various types of homology-cohomology theories are discussed. Our techniques can be used for producing natural elements in E-theory out of continuous fields with non-isomorphic fibers. An alternative definition for the Kasparov product in E-theory is proposed.