Negative K-theory of derived categories

We define negative K-groups for exact categories and for ``derived categories' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason. We prove localization and vanishing theorems for these groups. Dévissage (for noetherian abelian categories), additivity, and resolution hold. We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish. Our methods yield an explicit non-connective delooping of the K-theory of exact categories and chain complexes, generalizing constructions of Wagoner and Pedersen-Weibel. Extending a theorem of Auslander and Sherman, we discuss the K-theory homotopy fiber of ε^⊕→ ε and its implications for negative K-groups. In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

Separable <Emphasis Type="Italic">K</Emphasis>-linear categories

We define and investigate separable K-linear categories. We show that such a category C is locally finite and that every left C-module is projective. We apply our main results to characterize separable linear categories that are spanned by groupoids or delta categories.     

Wild ramification in number field extensions of prime degree

We show that if L/ K is a degree p extension of number fields which is wildly ramified at a prime ${\frak p}$ of K of residue characteristic p, then the ramification groups of ${\frak p}$ (in the splitting field of L over K) are uniquely determined by the ${\frak p}$-adic valuation of the discriminant of L /K.Received: 3 July 2002     

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FPR) is a class in the projective class group K₀(RΓ); the functorial Euler characteristic and functorial L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L²-Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

Local flat duality of abelian varieties

 Let K be a field complete for a discrete valuation and with algebraically closed residue field of positive characteristic p. We prove the existence of a non-degenerate pairing between the first (flat) cohomology group of an abelian variety A _K over K and the fundamental group of the Néron model of the dual abelian variety. This pairing extends to the p-primary components a pairing introduced by Shafarevich in [16]. We relate this pairing with Grothendieck's pairing. Received: 7 January 2002 / Revised version: 6 December 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14k05, 14F20, 14G22     

Algebraic K-theory and abstract homotopy theory

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer–Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.     

Équivalence rationnelle sur les hypersurfaces cubiques sur les corps <Emphasis Type="Italic">p</Emphasis>-adiques

 We study R-equivalence on cubic hypersurfaces, and explain how to construct families of rational curves. We show that for a smooth cubic hypersurface defined over a number field K, the Chow group of zero-cycles of degree 0 is trivial in almost every place of K. Received: 20 March 2002 Published online: 24 January 2003     

ℱ<Emphasis Type="Italic">K</Emphasis>-convex functions on metric spaces

 By an ℱK-convex function on a length metric space, we mean one that satisfies f ⁿ ≥ −Kf on all unitspeed geodesics. We show that natural ℱK-convex (-concave) functions occur in abundance on metric spaces of curvature bounded above (below) by K in the sense of Alexandrov. We prove Lipschitz extension and approximation theorems for ℱK-convex functions on CAT(K) spaces. Received: 10 May 2002 Mathematics Subject Classification (2000): 53C70, 52A41     

Sur les extensions purement inséparables

If K/k is a finite purely inseparable extension of fields, we are interested in the factorizations of K as a tensor product over k of intermediates fields of K/k. We introduce the notion of e-factorization that generalizes the notion of modular factorization. Contrary to modular factorization, K/k has always an e-factorization and its factors, when their number is maximum, are quasi-invariants.Received: 4 April 2002     

Countability via capacity

Abstract. Let K be a compact subset of {\bf C}, and let c denote logarithmic capacity. We prove that if and only if K is countable. As an application, we obtain a short proof of the scarcity theorem for countable analytic multifunctions. Received: 13 November 2000 / Published online: 18 January 2002     

K-Theory for Triangulated Categories 3 1/2 (A): A Detailed Proof of the Theorem of Homological Functors Dedicated to Daniel Quillen and to the Memory of Robert Thomason

Let A and B be Abelian categories. Let H: A B be a bounded -functor. We prove that H induces a natural map in higher K-theory. From a more precise analysis of the proof, we deduce that it is possible to define a K-theory of the bounded derived category of A, which contains Quillen's K-theory of A as a retract.     

Entropy and the combinatorial dimension

We solve Talagrand’s entropy problem: the L ₂-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley’s theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton’s Theorem and estimates on the uniform central limit theorem in the real valued case. Oblatum 10-XII-2001 & 4-IX-2002?Published online: 8 November 2002     

Orthogonal double covers by super‐extendable cycles

An Orthogonal Double Cover (ODC) of the complete graph K_n by an almost‐hamiltonian cycle is a decomposition of 2K_n into cycles of length n?1 such that the intersection of any two of them is exactly one edge. We introduce a new class of such decompositions. If n is a prime, the special structure of such a decomposition allows to expand it to an ODC of K_n+1 by an almost‐hamiltonian cycle. This yields the existence of an ODC of K_p+1 by an almost‐hamiltonian cycle for primes p of order 3 mod 4 and its eventual existence for arbitrary primes p. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 283–293, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10011     

Proper actions of lattices on contractible manifolds

Every lattice Γ in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if Γ acts on a contractible manifold W and if either?1) the action is properly discontinuous, or?2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1,?then dimW≥dimG/K. Oblatum 19-I-2001 & 24-IV-2002?Published online: 5 September 2002 RID="*" ID="*"The authors gratefully acknowledge support from the National Science Foundation.     

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes

We prove localization and Zariski-Mayer-Vietoris for higher Gro-thendieck-Witt groups, alias hermitian K-groups, of schemes admitting an ample family of line-bundles. No assumption on the characteristic is needed, and our schemes can be singular. Along the way, we prove Additivity, Fibration and Approximation theorems for the hermitian K-theory of exact categories with weak equivalences and duality.     

Double coverings of 2‐paths by Hamilton cycles*

A double Dudeney set in K_n is a multiset of Hamilton cycles in K_n having the property that each 2‐path in K_n lies in exactly two of the cycles. A double Dudeney set in K_n has been constructed when n ≥ 4 is even. In this paper, we construct a double Dudeney set in K_n when n ≥ 3 is odd. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 195–206, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ) DOI 10.1002/jcd.10003     

Dimension Theory and Nonstable K 1

This article provides an introduction to A. Bak's theory of group-valued functors on categories with structure and dimension and applies this theory to the algebraic K-theory functors nonstable K ₁ and K ₂.