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 ∊  ^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 ∊ ℤ, has only a finite number of isomorphism classes (see Huisgen-Zimmermann and Saorín, 2001).

Key Words:

• Artin algebra
• Exact structures
• Extensions
• Lift category
• Projective resolution

2000 Mathematics Subject Classification:

• 16G10
• 16E05
• 15A21

ACKNOWLEDGMENTS

Both authors thank the support of project “43374F” of Fondo Sectorial SEP-Conacyt.

Notes

Communicated by D. Happel.