On Modules and Complexes Without Self-Extensions

Abstract

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