Iyama's finiteness theorem via strongly quasi-hereditary algebras

作者：Ringel Claus Michael^*

来源：Journal of Pure and Applied Algebra, 2010, 214(9): 1687-1692.

DOI：10.1016/j.jpaa.2009.12.012

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.