Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite-dimensional algebras. This is the theory of τ-exceptional sequences, which was developed in joint work with Marsh, motivated by τ-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.
The interplay between theories for τ-rigid modules, torsion pairs, and wide subcategories is central to our discussions.
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