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Field-dependent homological behavior of finite dimensional algebras

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Abstract

It is shown that the little finitistic dimension of a finite dimensional algebra, i.e., the supremum of the finite projective dimensions attained on finitely generated modules, is not necessarily attained on a cyclic module. In general, arbitrarily high numbers of generators are required. Moreover, it is demonstrated that this phenomenon may depend on the base fieldk. In fact, for each integerd>-3, there exists a quiver Γ with a set ρ of paths such that the little finitistic dimension of the finite dimensional algebrakΓ/<ρ> is attained on a cyclic module precisely when |k|≥d. By contrast, the global dimension of finite dimensional monomial relation algebras does not depend on the base field.

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Authors and Affiliations

  1. Department of Mathematics, University of California, 93106, Santa Barbara, CA, USA

    Birge Zimmermann Huisgen

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  1. Birge Zimmermann Huisgen
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This research was partially supported by a grant from the National Science Foundation.

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Huisgen, B.Z. Field-dependent homological behavior of finite dimensional algebras. Manuscripta Math 82, 15–29 (1994). https://doi.org/10.1007/BF02567682

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  • DOI: https://doi.org/10.1007/BF02567682

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