Finitistic Dimensions of Artin Algebras with Two Simples and a Conjecture of Marczinzik

Abstract

Let Λ be an Artin algebra with a unique non-injective indecomposable projective module. In this situation, Marczinzik conjectured that the dominant dimension of Λ agrees with its finitistic dimension. In this paper, we give a proof of a stronger statement. As a byproduct, we obtain excellent control over the finitistic dimensions of Artin algebras with two simples and positive dominant dimension, and also establish the Gorenstein symmetry conjecture for all algebras under consideration.