Tilting modules over the truncated symmetric algebra (Q1320168)

In studying \(\text{mod }A\) for certain classes of finite dimensional algebras \(A\) it turned out that tilting modules play an important role since they relate the module category of the endomorphism ring \(B\) of a tilting module over \(A\) with \(\text{mod }A\). In fact, \(A\) and \(B\) have equivalent bounded derived categories. This allows a transfer of information between \(\text{mod }A\) and \(\text{mod }B\). Tilting modules over hereditary algebras of finite or tame representation type and their endomorphism rings are quite well understood, but little is known if \(A\) is a wild algebra, and almost nothing if \(A\) is wild and not hereditary. The truncated symmetric algebras are wild and not hereditary for \(n > 1\). It is the purpose of this article to determine all tilting modules \(M\) over the truncated symmetric algebra for \(n = 2\) which have the property that the endomorphism ring of each indecomposable direct summand of \(M\) is the ground field. We call these tilting modules exceptional. It is an open question whether all tilting modules over this specific algebra are exceptional.