Quasi-hereditary algebras related to local algebras

Abstract

For ⋀ a finite dimensional local algebra with radical N where Nⁿ = 0 ≠ N ^n-1 define (as a right A-module), then A(⋀) is quasi-hereditary and it has a unique heredity ideal J ₁(A).Assume ⋀ satisfies the right socle condition (the socle series and the radical series of ⋀_⋀ coincide). We show that then the algebra A_i/J₁ (A_i-1 ) is isomorphic to A _i-1 where A_i = A (⋀/Nⁱ ). for 1 ≤ i ≤ n. Moreover we determine the canonical module for A(⋀) and we show that the Ringel dual of A(⋀) is isomorphic to the endomorphism ring of as a left module, provided that ⋀ also satisfies the left socle condition.