OHIO UNIVERSITY CENTER FOR
RING THEORY AND ITS APPLICATIONS
LECTURE SERIES
Noncommutative Elementary Divisor Theory
by
Lawrence Levy
University of Nebraska
(Visiting CRA: April
28-May 3)
Lecture 1: Finitely generated modules over noncommutative PIDs
April 29, 2005 at Ohio State University
Lecture 2: Reduction to a direct-sum cancellation problem
April 30, 2005 at Ohio University
Lecture 3: Solution of the problem
May 1, 2005 at Ohio University
ABSTRACT
Commutative "Elementary Divisor Theory" states that, over any commutative
PID (principal ideal domain) R, every matrix A = diag(d_1,...,d_n) is equivalent
to a diagonal matrix --- under a transformation A --> PAQ with P and
Q invertible. Moreover, two matrices A and B of the same size are equivalent
if and only the direct sum of cyclic R-modules R/Rd_1 + ... + R/Rd_n is
isomorphic to the corresponding direct sum for the diagonal form of B.
A problem left over from the 1930's is to extend this to noncommutative
PIDs. It is easy to show that matrices can be diagonalized. The stumbling
block is the uniqueness theorem, which was known to be false, even for matrices
of rank 1.
A 1988 theorem of Guralnick, Levy, and Odenthal states that matrices
of rank 1 are the only exceptions. Otherwise the classical uniqueness theorem
holds. The solution transforms the original problem to a direct-sum cancellation
problem for modules over a different ring --- which is not a PID. The reason
that there are no exceptions in higher rank is given by a stability theorem
from K-theory.
These three talks will present the history of the problem and the main
parts of its solution, beginning with a summary of the structure of finitely
generated modules over noncommutative PIDs. A summary of the K-theory needed
at one point of the solution will be included.