Middle Terms of AR-sequences of Graded Kronecker Modules

Abstract

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