An introduction to tau-exceptional sequences

Joint work with Aslak Bakke Buan.

Exceptional sequences in module categories over hereditary algebras (e.g. path algebras of quivers) were introduced and studied by W. Crawley-Boevey and C. M. Ringel in the early 1990s, as a way of understanding the structure of such categories. They were motivated by the consideration of exceptional sequences in algebraic geometry by A. I. Bondal, A. L. Gorodontsev and A. N. Rudakov.

Exceptional sequences can also be considered over arbitrary finite dimensional algebras, but their behaviour is not so good in general: for example, complete exceptional sequences may not exist. We look at different ways of generalising to the hereditary case, with a focus on tau-exceptional sequences, recently introduced in joint work with A. B. Buan (NTNU), motivated by the tau-tilting theory of T. Adachi, O. Iyama and I. Reiten, and signed exceptional sequences in the hereditary case defined by K. Igusa and G. Todorov.