Regular stones of wild hereditary algebras (Q1321047)

Let \(A\) be a finite-dimensional wild hereditary algebra over an algebraically closed field \(k\). Recall that an indecomposable finitely generated \(A\)-module \(X\) is called a stone provided \(\text{Ext}^ 1_ A (X,X) = 0\). It is known that if \(A\) has more than two simple modules then there are infinitely many components in the Auslander-Reiten quiver of \(A\) containing stones. The authors prove that there are however only finitely many non-sincere regular components containing stones of quasi-length 2. Moreover, under the additional assumption that \(A\) has elementary modules with selfextensions it is shown that all stones in sincere regular components are quasi-simple. Here a regular module \(E\) is called elementary if it has no regular submodule \(U \neq 0\) with \(0 \neq E/U\) also regular.