Derived dimensions of representation-finite algebras

It is shown that the derived dimension of any representation-finite Artin algebra is at most one. Mathematics Subject Classification 2000: 16G10, 18E30, 18E10. Let T be a triangulated category, and I and J two full subcategories of T . Denote by 〈I〉 the smallest full subcategory of T containing I and closed under shifts, finite direct sums and direct summands. Denote by I∗J the full subcategory of T consisting of all the objects M ∈ T for which there exists a triangle I → M → J → I[1] with I ∈ I and J ∈ J . Put I ⋄ J = 〈I ∗ J 〉 and inductively 〈I〉n = { {0}, if n = 0; 〈I〉n−1 ⋄ 〈I〉, if n ≥ 1. The dimension of T is dimT := min{d|∃M ∈ T ,∋ T = 〈M〉d+1}, or ∞ if there is no such an M for any d. (ref. [9, Definition 3.2]). Let A be an abelian category. Then D(A), the derived category of bounded complexes over A, is a triangulated category. We call dimD(A) the derived dimension of A. Let A be an associative algebra with identity, and A-mod the category of finitely generated left A-modules. Then we also say dimD(A−mod) is the derived dimension of A. (ref. [3]). The derived dimension of an Artin algebra is closed related to its Loewy length, global dimension, and representation dimension. Especially, it provides a lower ∗The author is supported by NSFC (Project 10731070).