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A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras

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Abstract

Given a finite-dimensional algebra Λ, we show that a frequently satisfied finiteness condition for the category \(P^\infty\) Λ-mod) of all finitely generated (left) Λ-modules of finite projective dimension,namely contravariant finiteness of \(P^\infty\) (Λ-mod) in Λ-mod, forces arbitrary modules of finite projective dimension to be direct limits of objects in \(P^\infty\) (Λ-mod). Among numerous applications, this yields an encompassing sufficient condition for the validity of the first finitistic dimensionconjecture, that is, for the little finitistic dimension of Λ to coincide with the big (this is well known to fail overfinite-dimensional algebras in general).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, U.S.A.

    B. Huisgen-Zimmermann

  2. Department of Mathematics, The Norwegian University for Science and Technology, 7055, Dragvoll, Norway

    S. O. SmalØ

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  1. B. Huisgen-Zimmermann
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  2. S. O. SmalØ
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Huisgen-Zimmermann, B., SmalØ, S.O. A Homological Bridge Between Finite and Infinite-Dimensional Representations of Algebras. Algebras and Representation Theory 1, 169–188 (1998). https://doi.org/10.1023/A:1009948721602

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