On the computation of a-invariants


Thea-invariant of a graded Cohen-Macaulay ring is the least degree of a generator of its graded canonical module. We compute thea-invariants of (i) graded algebras with straightening laws on upper semi-modular lattices and (ii) the Stanley-Reisner rings of shellable weighted simplicial complexes. The formulas obtained are applied to rings defined by determinantal and pfaffian ideals.

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  1. 1.

    W. Bruns, W. and Vetter, U.: Determinantal rings. Lect. Notes Math.1327, Springer 1988

  2. 2.

    De Concini, C., Eisenbud, D., and Procesi, C.: Hodge algebras. Astérisque91, 1982

  3. 3.

    Eisenbud, D.: Introduction to algebras with straightening laws. In: McDonald, B. R. (Ed.), Ring theory and algebra III, M. Dekker, 1980, pages 243–267

  4. 4.

    Flenner, H.: Quasi-homogene rationale Singularitäten. Arch. Math.36, 35–44 (1981)

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Gräbe, H.-G.: Streckungsringe. Dissertation B, Erfurt/Mühlhausen, 1988

  6. 6.

    Herzog, J. and Ngo Viet Trung: Gröbner bases and multiplicity of determinantal and pfaffian ideals, to appear in Adv. Math

  7. 7.

    Kleppe, H. and Laksov, D.: The algebraic structure and deformation of Pfaffian schemes. J. Algebra64, 167–189 (1980)

    MATH  Article  MathSciNet  Google Scholar 

  8. 8.

    Stanley, R. P.: Combinatorics and commutative algebra. Birkhäuser, 1983

  9. 9.

    Yoshino, Y.: The canonical modules of graded rings defined by generic matrices. Nagoya Math. J.81, 105–112 (1981)

    MATH  MathSciNet  Google Scholar 

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Bruns, W., Herzog, J. On the computation of a-invariants. Manuscripta Math 77, 201–213 (1992). https://doi.org/10.1007/BF02567054

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  • Partial Order
  • Simplicial Complex
  • Minimal Element
  • Hilbert Series
  • Canonical Module