Middle Terms of AR-sequences of Graded Kronecker Modules

Abstract

Let \((T(n),\Omega )\) be the covering of the generalized Kronecker quiver K(n), where \(\Omega \) is a bipartite orientation. Then there exists a reflection functor \(\sigma \) on the category \({{\,\textrm{mod}\,}}(T(n),\Omega )\). Suppose that \(0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\) is an AR-sequence in the regular component \(\mathcal {D}\) of \({{\,\textrm{mod}\,}}(T(n),\Omega )\), and b(Z) is the number of flow modules in the \(\sigma \)-orbit of Z. Then the middle term Y is a sink (source or flow) module if and only if \(\sigma Z\) is a sink (source or flow) module. Moreover, their radii and centers satisfy \(r(Y)=r(\sigma Z)+1\) and \(C(Y)=C(\sigma Z)\).