On the vanishing of negative K-groups

Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel.Comment: Math. Ann. (to appear

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

Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. 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 a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.Comment: Revised and expanded version, 42 pages

The K-theoretic Farrell-Jones Conjecture for hyperbolic groups

We prove the K-theoretic Farrell-Jones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.Comment: 33 pages; final version; to appear in Invent. Mat

A Homology Theory for Hybrid Systems: Hybrid Homology

Abstract. By transferring the theory of hybrid systems to a categorical framework, it is possible to develop a homology theory for hybrid systems: hybrid homology. This is achieved by considering the underlying “space ” of a hybrid system—its hybrid space or H-space. The homotopy colimit can be applied to this H-space to obtain a single topological space; the hybrid homology of an H-space is the homology of this space. The result is a spectral sequence converging to the hybrid homology of an H-space, providing a concrete way to compute this homology. Moreover, the hybrid homology of the H-space underlying a hybrid system gives useful information about the behavior of this system: the vanishing of the first hybrid homology of this H-space—when it is contractible and finite—implies that this hybrid system is not Zeno.