The Support τ-tilting Complements To τ-rigid Modules

Tilting theory plays an important part in the representation theory of algebra. As a generalization of Morita equivalence, it studies the equivalence between tilting torsion pairs in two module categories. Recently, Adachi, Iya-ma and Reiten introduce a generalization of classical tilting theory, which is called r-tilting theory. By using this theory, they establish a bijection between cluster tilting objects and a kind of module which they call support τ tilting module in the module category of cluster tilted algberas, and further explain the relation between the cluster category and module category of cluster tilted algebra.In tilting theory, the number of tilting complements to partial tilting modules has been the focus of study all the time.Naturally, we hope this new theory can present us a solution to this problem. Because the sup-port τ-tilting modules coincide with the support tilting modules in the mod-ule category of a hereditary algebra, we calculate the number of support τ-tilting complements to τ-rigid modules by virtue of the results introduced by Obaid, Nauman,Fakieh and Ringel.One of the limitations in classical tilting theory is that the mutation of tilt-ing modules is not always possible, and thus the tilting quiver is not connected all the time. In the last chapter of this thesis, we use τ-tilting theory to prove that the tilting quiver is connected if there exists a faithful projective-injective module in modA.