On the global and \nabla-filtration dimensions of quasi-hereditary algebras

Abstract

In this paper we consider how the \nabla-, \Delta- and global dimensions of a quasi-hereditary algebra are interrelated. We first consider quasi-hereditary algebras with simple preserving duality and such that if \mu < \lambda then \nabla fd(L(\mu)) < \nabla fd(L(\lambda)) where \mu, \lambda are in the poset and L(\mu), L(\lambda) are the corresponding simples. We show that in this case the global dimension of the algebra is twice its \nabla-filtration dimension. We then consider more general quasi-hereditary algebras and look at how these dimensions are affected by the Ringel dual and by two forms of truncation. We restrict again to quasi-hereditary algebras with simple preserving duality and consider various orders on the poset compatible with quasi-hereditary structure and the \nabla-, \Delta- and injective dimensions of the simple and the costandard modules.