An effective criterion for uniqueness of the tilting module (Q1206079)

Let \(A\) be a finite dimensional algebra over an algebraically closed field \(k\) and \(M\) a finite dimensional left \(A\)-module. If proj.dim. \(M\leq 1\), \(\text{Ext}^ 1_ A(M,M) = 0\) and the number of non- isomorphic indecomposable direct summands of \(M\) is equal to that of non- isomorphic simple modules, then \(M\) is called a tilting module. Of course, \(_ AA\) is a tilting module. In the paper the authors give conditions to guarantee that \(_ AA\) is the unique tilting module.