International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 39, Pages 2065-2084
Moduli space of filtered -ringstructures over a filtered ring
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana 61801, IL, USA
Received 10 April 2003
Copyright © 2004 Donald Yau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered -ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings , where is between and , with the -adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered -ring structures over is canonically isomorphic to the set of ring maps from some universal ring to . From a local perspective, we demonstrate the existence of uncountably many mutually nonisomorphic filtered -ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over -algebras.