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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Formulas for the multiplicity of graded algebras
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by Yu Xie PDF
Trans. Amer. Math. Soc. 364 (2012), 4085-4106 Request permission

Abstract:

Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along $\textrm {Proj} (B)$. Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier’s Plücker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.
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Additional Information
  • Yu Xie
  • Affiliation: Department of Mathematics, The University of Notre Dame, South Bend, Indiana 46556
  • Email: yxie@nd.edu
  • Received by editor(s): October 7, 2009
  • Received by editor(s) in revised form: July 26, 2010
  • Published electronically: March 28, 2012
  • Additional Notes: This paper is based on the author’s Ph.D. thesis, written under the direction of Professor Bernd Ulrich. The author sincerely thanks Professor Ulrich for suggesting the problem and for advice and many helpful discussions.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 4085-4106
  • MSC (2010): Primary 13H15, 13A30; Secondary 14J70, 14B05
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05434-1
  • MathSciNet review: 2912446