Higher Class Groups of Orders and Integral Group Rings (original) (raw)

The Euler class groups of polynomial rings and unimodular elements in projective modules

Journal of Pure and Applied Algebra, 2003

Let A be a Noetherian ring of Krull dimension n containing the ÿeld of rationals. Let P be a projective A[T ]-module of rank n with trivial determinant such that the A-module P=TP has a free summand of rank one. It is proved that if n is even, then P has a free summand of rank one if it maps onto an ideal I of A[T ] of height n which is generated by n elements.

K- and L-theory of Group Rings

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) - Vol. I: Plenary Lectures and Ceremonies, Vols. II-IV: Invited Lectures, 2010

This article will explore the K-and L-theory of group rings and their applications to algebra, geometry and topology. The Farrell-Jones Conjecture characterizes K-and L-theory groups. It has many implications, including the Borel and Novikov Conjectures for topological rigidity. Its current status, and many of its consequences are surveyed. . Primary 18F25; Secondary 57XX.

Grothendieck Groups of Invariant Rings: Filtrations

Proceedings of the London Mathematical Society, 1993

We investigate the Grothendieck group G a (R) of finitely generated modules over the ring of invariants R = S G of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G 0 (R) that is defined in terms of (Gabriel-Rentschler) Krull dimension, properties of G 0 (R) are derived to a large extent from the connections between the sets of prime ideals of 5 and R. A crucial ingredient is an equivalence relation ~ on Spec/? that was introduced by Montgomery [25]. For example, we show that

On Central Ω-Krull Rings and their Class Groups

Canadian Journal of Mathematics, 1984

The aim of this note is to study the class group of a central fi-Krull ring and to determine in some cases whether a twisted (semi) group ring is a central Q-Krull ring. In [8] we defined an fi-Krull ring as a generalization of a commutative Krull domain. In the commutative theory, the class group plays an important role. In the second and third section, we generalize some results to the noncommutative case, in particular the relation between the class group of a central fi-Krull ring and the class group of a localization. Some results are obtained in case the ring is graded. Theorem 3.2 establishes the relation between the class group and the graded class group. In particular, in the P.I. case we obtain that the class group is equal to the graded class group. As a consequence of a result on direct limits of fi-Krull rings, we are able to derive a necessary and sufficient condition in order that a polynomial ring R[ (Xj) iOEl ] (I may be infinite) is a central 12-Krull ring. We also have RRR CI OR) = Cl(tf[(^) zG/ ]).