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Bounds for multiplicities

Authors: Jürgen Herzog and Hema Srinivasan
Journal: Trans. Amer. Math. Soc. 350 (1998), 2879-2902
MSC (1991): Primary 13H15, 13D25, 13Xxx
MathSciNet review: 1458304
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Abstract: 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

Hema Srinivasan
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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.
Article copyright: © Copyright 1998 American Mathematical Society

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