## Formulas for the multiplicity of graded algebras

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## 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 Teissierâ€™s PlĂĽcker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.## References

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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 authorâ€™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