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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Bounds for multiplicities
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by Jürgen Herzog and Hema Srinivasan PDF
Trans. Amer. Math. Soc. 350 (1998), 2879-2902 Request permission


Let $R=K[x_1,x_2,\ldots , x_n]$ and $S=R/I$ be a homogeneous $K$-algebra. We establish bounds for the multiplicity of certain homogeneous $K$-algebras $S$ in terms of the shifts in a free resolution of $S$ over $R$. Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasi-pure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals $I$. As a consequence, we get a bound for the regularity of $S$. Further, when $S$ is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a $p$-linear resolution.
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Additional Information
  • Jürgen Herzog
  • Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen, Postfach 103764, Essen 45117, Germany
  • MR Author ID: 189999
  • Email:
  • Hema Srinivasan
  • Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
  • MR Author ID: 269661
  • ORCID: 0000-0001-7509-8194
  • Email:
  • Received by editor(s): July 4, 1996
  • Additional Notes: The second author was supported in part by grants from National Science Foundation and U.M. Research Board.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 2879-2902
  • MSC (1991): Primary 13H15, 13D25, 13Xxx
  • DOI:
  • MathSciNet review: 1458304