Abstract
We prove a theorem which gives a bijection between the support tau -tilting modules over a given finite-dimensional algebra A and the support tau -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are tau -tilting-finite wild blocks with more than one simple module. We then go on to classify all support tau -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all tau -rigid modules over (not necessarily symmetric) string algebras.
Highlights
The theory of support τ -tilting modules, as introduced by Adachi, Iyama and Reiten in [4], is related to, and to some extent generalizes, several classical concepts in the representation theory of finite dimensional algebras.On the one hand, it is related to silting theory for triangulated categories, which was introduced by Keller and Vossieck in [23] and provides a generalization of tilting theory
It is related to silting theory for triangulated categories, which was introduced by Keller and Vossieck in [23] and provides a generalization of tilting theory
Τ -tilting theory is related to mutation theory, which has its origins in the Bernstein–Gelfand–Ponomarev reflection functors
Summary
The theory of support τ -tilting modules, as introduced by Adachi, Iyama and Reiten in [4], is related to, and to some extent generalizes, several classical concepts in the representation theory of finite dimensional algebras. We will be concerned with determining all basic two-term silting complexes (or equivalently support τ -tilting modules) for various finite dimensional algebras A defined over an algebraically closed field To this end, we prove the following very general reduction theorem: Theorem 1 (see Theorem 11) For an ideal I which is generated by central elements and contained in the Jacobson radical of A, the g-vectors of indecomposable τ -rigid (respectively support τ -tilting) modules over A coincide with the ones for A/I , as do the mutation quivers. Using a result of Aihara and Mizuno [7], we deduce the following theorem: Theorem 3 All tilting complexes over an algebra of dihedral, semidihedral or quaternion type can be obtained from A (as a module over itself) by iterated tilting mutation.
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