The Hochschild-Kostant-Rosenberg theorem, which relates the Hochschild homology of an algebra to its modules of differential $n$-forms, can be considered a benchmark for smoothness of an algebra. This notion is used here in the search for a conception of smoothness formulated in the context of codifferential categories, both commutative and noncommutative. Since it is desirable to adapt such a conception to the noncommutative context, the theory of this domain is developed considerably; a significant result in this direction establishes a connection between noncommutative codifferential categories and commutative ones.
This investigation necessitates, then, both the formulation of a notion of smoothness for $T$-algebras in codifferential categories and an adaptation of the Hochschild-Kostant-Rosenberg theorem to a wide variety of contexts which includes noncommutative ones. The former consideration fosters both the notion of a smooth monad, and a formulation of Andr\'e-Quillen homology in codifferential categories; the latter engenders a highly adaptable version of the Hochschild-Kostant-Rosenberg theorem. Specifically, it is shown that for any algebra modality there exists a corresponding Hochschild-Kostant-Rosenberg theorem. This includes a version of the theorem for the free associative algebra monad, the conclusion of which is satisfied by noncommutative smooth algebras.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/36703 |
| Date | January 2017 |
| Creators | O'Neill, Keith |
| Contributors | Blute, Richard |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
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