Tau-tilting Theory in Representation Theory of Finite Dimensional Algebras
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Tau-tilting theory was recently introduced by Adachi, Iyama and Reiten. Their main aim was to develop a generalization of classical tilting theory where mutation is always possible. The inspiration for this came mainly from the recently developed cluster-tilting theory where there is such a result. An inspiration for using tau-rigid modules, which were introduced by Auslander and Smalø in the early eighties and are generalizations of classical partial tilting modules, also came from cluster-tilting theory where the notion of tau-rigid appears naturally in connection with modules over 2-CY-tilted algebras. In order for mutation to be always possible one needs also take into account the notion of support tilting as introduced by Ingalls/ Thomas and Ringel. In this way we get that an almost complete support tau-tilting module (or to be exact, an almost complete support tau-tilting pair) over any finite dimensional algebra has two complements, i.e. mutation is always possible.