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Predicting syzygies over monomial relations algebras

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Abstract

We show that all projective resolutions over a monomial relations algebra Λ simplify drastically at the stage of the second syzygy; more precisely, we show that the kernel of any homomorphism between two projective left Λ-modules is isomorphic to a direct sum of principal left ideals generated by paths. As consequences, we obtain:

  1. (a)

    a tight approximation of the finitistic dimensions of Λ in terms of the (very accessible) projective dimensions of the principal left ideals generated by paths;

  2. (b)

    a basis for comparison of the ‘big’ and ‘little’ finitistic dimensions of Λ, yielding in particular that these two invariants cannot differ by more than 1 and that they are equal in ‘most’ cases;

  3. (c)

    manageable algorithms for computation of finitistic dimensions.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, 84112, Salt Lake City, Utah, 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. Predicting syzygies over monomial relations algebras. Manuscripta Math 70, 157–182 (1991). https://doi.org/10.1007/BF02568368

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  • DOI: https://doi.org/10.1007/BF02568368

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