Symmetries and Connected Components of the AR-quiver of a Gorenstein Local Ring

Abstract

Let $(A,\mathfrak {m})$ be a commutative complete equi-characteristic Gorenstein isolated singularity of dimension d with $k = A/\mathfrak {m}$ algebraically closed. Let Γ(A) be the AR (Auslander-Reiten) quiver of A. Let $\mathcal {P}$ be a property of maximal Cohen-Macaulay A-modules. We show that some naturally defined properties $\mathcal {P}$ define a union of connected components of Γ(A). So in this case if there is a maximal Cohen-Macaulay module satisfying $\mathcal {P}$ and if A is not of finite representation type then there exists a family {M_n}_n≥ 1 of maximal Cohen-Macaulay indecomposable modules satisfying $\mathcal {P}$ with multiplicity e(M_n) > n. Let $\underline {\Gamma }(A)$ be the stable quiver. We show that there are many symmetries in $\underline {\Gamma }(A)$. As an application we show that if $(A,\mathfrak {m})$ is a two dimensional Gorenstein isolated singularity with multiplicity e(A) ≥ 3 then for all n ≥ 1 there exists an indecomposable self-dual maximal Cohen-Macaulay A-module of rank n.