Resumen

Finite dimensional algebras are often given as path algebras modulo an ideal I of relations. When I is trivial, the algebra is so-called hereditary, and the representation theory of such algebras is well understood. For such algebras, an important tool is the notion of exceptional sequences, which provides a technique to identify descending sequences of module categories (= categories of representations) of decreasing rank, sitting inside a given module category as wide subcategories. “Wide” means: kernels, cokernels and exact sequences should be preserved.

For a non-trivial ideal I, the classical theory of exceptional sequences is often not very helpful, since most algebras do not admit such sequences. In a series of papers with Eric J. Hanson and/or Bethany R. Marsh, we have developed a theory of tau-exceptional sequences, which is a natural generalization to this more general setting. I will explain some of the main ideas, features and shortcomings of this theory and illustrate a mutation operation for tau-exceptional sequences. This was introduced in a recent preprint, arxiv: 2402.10301.