On the Global and ∇-filtration Dimensions of Quasi-hereditary Algebras

In this paper we consider how the∇–, ∆– 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 < λ then ∇. f.d.(L(μ)) < ∇. f.d.(L(λ)) where μ, λ are 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 ∇-, ∆and injective dimensions of the simple and the costandard modules.