On the global and <f>∇</f>-filtration dimensions of quasi-hereditary algebras

In this paper, we consider how the <f>∇</f>-, <f>Δ</f>- and global dimensions of a quasi-hereditary algebra are interrelated. We first consider quasi-hereditary algebras with simple preserving duality and such that if <f>μ<λ</f> then <f>∇.f.d.(L(μ))<∇.f.d.(L(λ))</f>, where <f>μ, λ</f> are in the poset and <f>L(μ), L(λ)</f> are the corresponding simples. We show that in this case the global dimension of the algebra is twice its <f>∇</f>-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 <f>∇</f>-, <f>Δ</f>- and injective dimensions of the simple and the costandard modules. [Copyright &y& Elsevier]