Universal derivations and universal ring constructions

Abstract

If a K-ring S is constructed from a K-ring R by adjoining certain new generators and relations, then the S-bimodule ΩK(S) with a universal K-derivation d: S →ΩK(S) can be constructed from the corresponding R-bimodule ΩK(R) by extending scalars to S, and adjoining formal derivatives of the new generators and relations. By studying this bimodule it is shown that a large number of natural universal constructions preserve the class of right hereditary K-rings (K semisimple Artinian), including the constructions of universal localization (which had resisted earlier techniques) and certain direct limits of known constructions. The same technique gives information on Euler characteristics of modules (Lewin-Schreier formulas). To study universal localizations of a ring R which may not contain a semisimple Artin ring K, a different technique is used. © 1978, University of California, Berkeley. All Rights Reserved.