Papers / Preprints. The papers below may differ slightly from the final published versions.
- Periodic algebras. A finite-dimensional algebra is periodic if it has a periodic projective resolution as a module over its enveloping algebra. Classical examples include the preprojective algebras of (simply-laced) Dynkin graphs. Many additional examples have been given by Bialkowski-Erdmann-Skowronski and by Brenner-Butler-King, but a classification of periodic algebras seems difficult. In particular, it would be nice to know whether deformations of periodic algebras are necessarily periodic.
- Periodic resolutions and
self-injective algebras of finite type. Preprint
(2008).
We show that the class of periodic algebras is closed under Galois covers, and apply this result to deduce that all self-injective algebras of finite representation type are periodic. This answers a question of Butler's. Moreover, by work of Green, Snashall and Solberg it shows that the Hochschild cohomology ring, modulo nilpotent elements, of a finite-type self-injective algebra is isomorphic to k[x]. We also calculate the periods of most finite-type self-injective algebras.
- Stable Calabi-Yau dimensions of self-injective algebras of finite type. In preparation.
- Stable equivalence of finite-dimensional algebras. Loosely speaking, two algebras over a field k are said to be stably equivalent if their module categories are equivalent modulo projectives. Auslander conjectured that two stably equivalent algebras have the same number of nonprojective simple modules (up to isomorphism). While this conjecture has been verified in a few important cases, it remains unresolved in general and motivates much of the research in this area.
- Finitistic dimension and modules of finite projective dimension.
- Noncommutative Rings.
