HIGH ORDER KÄHLER MODULES OF NONCOMMUTATIVE RING EXTENSIONS

Abstract

We construct the high order Kähler modules of noncommutative ring extensions B/A and show their fundamental properties. Our Kähler modules represent not only high order left derivations for one-sided modules but also high order central derivations for bimodules, which are usual derivations. This new viewpoint enables us to prove new results which were not known even though B is an algebra over a commutative ring A. Our results are the decomposition of Kähler modules by an idempotent element, exact sequences of Kähler modules, the Kähler modules of factor rings, and the relation to separable extensions. In particular, our exact sequences of high order Kähler modules were not known even though B is commutative.

ACKNOWLEDGMENT

This research was partially supported by the Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.