On modules and complexes without self-extensions

Abstract

Let A be an Artin algebra over a commutative Artinian ring k. If M is a finitely generated left A-module, let f:PM→M denote the minimal projective cover of M and let Ω(M)=kerf. In this paper the following interesting result is proven: if M,N are finitely generated left A-modules such that Ext1A(M,M)=Ext1A(N,N)=0, then M≅N if and only if M/radM≅N/radN and Ω(M)≅Ω(N). The proof is not easy and depends on obtaining results about "lift categories'' and "categories of complexes''. A corollary of the above result is obtained. In proving these two results, the authors also obtain a result which is very closely related to Corollary 9 of [B. Huisgen-Zimmermann and M. Saorín Castaño, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4757–4777