On Modules and Complexes Without Self-Extensions

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?? is a sequence of nonnegative integers almost all of them zero, then we prove that the family of objects X ?  ^b (Λ), the bounded derived category of Λ, with Hom ^b (Λ)(X,X[1]) = 0 and dim _k H ⁱ (X) = d _i for all i ? ?, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001 Huisgen-Zimmermann , B. , Saorín , M. ( 2001 ). Geometry of chain complexes and outer automorphisms under derived equivalence . Trans. Amer. Math. Soc. 353 : 4757 – 4777 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]).