Christian Haesemeyer

Descent Properties of Homotopy K-Theory

In this paper, we show that the widely held expectation that Weibel’s homotopy K theory satisfies cdh-descent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequences are derived. Finally, some evidence for a conjecture of Weibel concerning negative K -theory is given.

K-regularity, cdh-fibrant Hochschild homology, and a conjecture of Vorst

It is a well-known fact that algebraic if-theory is homotopy invariant as a functor on regular schemes; if X is a regular scheme, then the natural map Kn(X) ? Kn(X x ?1) is an isomorphism for all n eZ. This is false in general for nonregular schemes and rings. To express this failure, Bass introduced the terminology that, for any contravari ant functor V defined on schemes, a scheme X is called V-regular if the pullback maps V(X) ?> V(X x Ar) are isomorphisms for all r > 0. If X = Spec(i?), we also say that R is P-regular. Thus regular schemes are ifn-regular for every n. In contrast, it was observed as long ago as in [2] that a nonreduced affine scheme can never be K\-regular. In particular, if A is an Artinian ring (that is, a O-dimensional Noetherian ring), then A is regular (that is, reduced) if and only if A is ifi-regular. In [17], Vorst conjectured that for an affine scheme X, of finite type over a field F and of dimension d, regularity and K?+i-regularity are equivalent; Vorst proved this conjecture for d = 1 (by proving that ^-regularity implies normality). In this paper, we prove Vorsts conjecture in all dimensions provided the char acteristic of the ground field F is zero. In fact we prove a stronger statement. We say that X is regular in codimension n in X. Note that for all n Z, if a ring R is ifn-regular, then it is Kn-i-regular. This is proved in [17] for n > 1 and in [6, 4.4] for n KH(X), where K(X) is the algebraic if-theory spectrum of X and KH(X) is the homotopy if-theory of X defined in [19]. We write FK(R) for fK(Spec(R)). Theorem 0.1. Let R be a commutative ring which is essentially of finite type over a field F of characteristic 0. Then the following hold. (a) If Tk(R) is n-connected, then R is regular in codimension KHi(X) is an isomorphism for i < n and a surjection for i = n +1, so that Fk(X)

Norm Varieties and the Chain Lemma (After Markus Rost)

The goal of this paper is to present proofs of two results of Markus Rost, the Chain Lemma 1 and the Norm Principle 3. These are the steps needed to complete the published verification of the Bloch– Kato conjecture, that the norm residue maps are isomorphisms for every prime p, every n and every field k containing 1/ p.

The

Recent advances in computational techniques for K-theory allow us to describe the K-theory of toric varieties in terms of the K-theory of fields and simple cohomological data.

K

We show that if X is a toric scheme over a regular ring containing a field then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was known in characteristic 0. The affine case of our result was conjectured by Gubeladze.

-theory of toric varieties

This meeting brought together algebraic geometers, algebraic topologists and geometric topologists, all of whom use algebraic K-theory. The talks and discussions involved all the participants. Mathematics Subject Classification (2000): 19xx. Introduction by the Organisers There have been dramatic advances in algebraic K-theory recently, especially in the computation and understanding of negative K-groups and of nilpotent phenomena in algebraic K-theory. Parallel advances have used remarkably different methods. Quite complete computations for the algebraic K-theory of commutative algebras over fields have been obtained using algebraic geometric techniques. On the other hand, the Farrell-Jones conjecture implies results on the K-theory for arbitrary rings. Proofs here use controlled topology and differential geometry. Given the diversity of interests and backgrounds of the 28 participants in our mini-workshop, we encouraged everyone to make their talks accessible to a wide audience and scheduled five expository talks. The opening talk of the conference was an inspiring talk by Charles Weibel, on the work of Daniel Quillen, the creator of higher algebraic K-theory, who died at the end of April. Wolfgang Luck spoke on the Farrell-Jones conjecture. Jim Davis applied the Farrell-Jones conjecture to give a foundational result on algebraic K-theory, showing that geometric techniques have algebraic consequences for the iterated NK-groups. Bjorn Dundas gave a survey of trace methods on algebraic K-theory, focusing on topological 1470 Oberwolfach Report 27/2011 cyclic homology and his new integral homotopy cartesian square. Christian Haesemeyer gave a survey of algebraic K-theory of singularities and new techniques for computing negative K-theory and NK-theory for commutative Q-algebras. The idea of the expository talks worked quite well; it was remarkable how many of the speakers relied on them. The mini-workshop had a full schedule; in addition to the five expository talks there were seventeen research talks. There were computational talks (Teena Gerhardt, Charles Weibel, Daniel-Juan Pineda), foundational talks (Bruce Williams, Lars Hesselholt, Max Karoubi, Guilermo Cortinas, Jens Hornbostel, Andrew Blumberg, Thomas Geisser), applications of ideas from K-theory to geometric topology (Ib Madsen, Frank Connolly, Qayum Khan, Ian Hambleton, Michael Weiss, Wolfgang Steimle), as well as the proof of the Farrell-Jones Conjecture for the group SLn(Z) (Holger Reich). The talk of Charles Weibel was notable since the topic was research done at the workshop. Weibel’s talk connected and compared two different computations of the NKqR groups, one done by algebraic geometry and one done by geometric topology. This was emblematic of a successful implementation of the original goal of the workshop to compare and contrast two powerful but quite distinct approaches to algebraic K-theory.