Smoothness and rigidity of tensor algebras and their factors

Abstract

In this paper, we begin a study of the lower dimensional relative cohomology groups of algebras defined by Hochschild [7]. Section 1 gives a few known but basic results. The results show how the relative cohomology groups for noncommutative rings of dimensions 1,2, and 3 correspond to the cohomology groups studied in commutative deformation theory. We show, for example, that they are the cohomology groups of a noncommutative cotangent complex. This correspondence motivates our definitions of smoothness and rigidity. From the results of Section 1, we see that if R is a semisimple ring and A is an augmented ring over R, i.e., there are ring morphisms R -+ A -R whose composition is the identity, then A is smooth over R if it is a tensor algebra over R. In Section 2, we show that the converse is also true under some additional hypothesis on A. For the remainder of the paper, we only consider tensor algebras that are associated to K-species, where K is a field (see Section 3 for definitions). Section 4 gives results on the rigidity of factors of these tensor algebras. It is shown that there are tensor algebras, each of whose factor rings is rigid. We call such tensor algebras factor rigid and classify them in Section 5. Note that the phenomenon of factor rigidity does not occur in the commutative theory, where a polynomial ring over a field K is factor rigid over k if and only if it is just the field k itself. I would like to thank D.S. Rim for introducing me to this subject and for his many helpful suggestions.