Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order. Received August 1, 1998     

The Leibniz formula in algebraic K-theory

The paper may be viewed as an addendum to a paper of Thomason and Throbaugh, where the K-theory of algebraic varieties is equipped with relative K-groups. It is proved that this enriched K-theory satisfies the Panin—Smirnov axioms for ring cohomology theories of algebraic varieties. In particular, it is proved that the Leibniz formula, describing the interaction between multiplication and differential, holds in this case. The language of symmetric spectra and of monoidal model categories is used. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 264–292.     

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”.     

On products in algebraic K-theory

This paper investigates the product structure in algebraic K-theory of rings. The first objective is to understand the relationships between products and the kernel of the Hurewicz homomorphism relating the algebraic K-theory of any ring to the integral homology of its linear groups. The second part of the paper is devoted to the ring of integers . Using recent results of V. Voevodsky we completely determine the products in tensored with the ring of 2-adic integers. Received: January 3, 1999.     

The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H ⁱ (k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K _n ^M (k)?K _n (k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K _n ^M (k) and K _n (k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K _n (X,ℤ/p ^r )=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture. Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999     

On algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S¹-action so that the quotient Y=X/S¹ is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ,

Full-size image

is onto, where , denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that

Full-size image

for any real algebraic variety X. As an application we will show that for a compact connected Lie group G .     

Equivariant K-theory of compactifications of algebraic groups

In this paper we describe the G × G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl group. We further give an explicit basis of K(X) over K(G/B), and also determine the structure constants with respect to this basis.     

Higher algebraic K-theory for actions of diagonalizable groups

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type. Mathematics Subject Classification (2000) 19E08, 14L30