Type: Article

On modules and complexes without self-extensions

Raymundo Bautista; Efrén Pérez;

Abstract:

Let Lambda be an Artin algebra over a commutative Artinian ring, k. If M is a finitely generated left Lambda-module, we denote by Omega(M) the kernel of eta(M) : PM -> M a minimal projective cover. We prove that if M and N are finitely generated left Lambda-modules and Ext(Lambda)(1) (M, M) = 0, Ext(Lambda)(1) (N, N) = 0, then M congruent to N if and only if M/radM congruent to N/rad N and Omega(M) congruent to Omega(N). Now if k is an algebraically closed field and (d(i))(i epsilon Z) is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X epsilon D(b)(Lambda), the bounded derived category of Lambda, with Hom(D)b((Lambda)) (X, X[1]) = 0 and dim(k)H(i)(X) = d(i) for all i epsilon Z, has only a finite number of isomorphism classes ( see Huisgen-Zimmermann and Saorin, 2001).