Iyama's finiteness theorem via strongly quasi-hereditary algebras

Ringel CM. Iyama's finiteness theorem via strongly quasi-hereditary algebras. Journal of Pure and Applied Algebra. 2010;214(9):1687-1692.Let Lambda be an artin algebra and X a finitely generated Lambda-module. Iyama has shown that there exists a Module Y such that the endomorphism ring Gamma of X circle times Y is quasi-hereditary, with a heredity chain of length n, and that the global dimension of Gamma is bounded by this n. In general, one only knows that a quasi-hereditary algebra with a heredity chain of length It must have global dimension at most 2n - 2. We want to show that Iyama's better bound is related to the fact that the ring Gamma lie constructs is not only quasi-hereditary, but even left strongly quasi-hereditary. By definition, the left strongly quasi-hereditary algebras are the quasi-hereditary algebras with all standard left modules of projective dimension at most 1. (C) 2009 Elsevier B.V. All rights reserved