2005/08/09 by Karin Erdmann, Alison E. Parker
Mathematics · #math.RT #math.RA #msc:16G99
published as J. Pure and Appl. Algebra (2004), 194, 95-111 · 18 pages, uses xypic
arxiv created 2005/08/09 · arxiv updated 2009/12/01
In this paper we consider how the ∇-, Δ- and global dimensions of a quasi-hereditary algebra are interrelated. We first consider quasi-hereditary algebras with simple preserving duality and such that if μ< λthen ∇ fd(L(μ)) < ∇ fd(L(λ)) where μ, λare in the poset and L(μ), L(λ) are the corresponding simples. We show that in this case the global dimension of the algebra is twice its ∇-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 ∇-, Δ- and injective dimensions of the simple and the costandard modules.