### 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 ⊆ 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 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 Pl̈ucker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

Original language | English (US) |
---|---|

Pages (from-to) | 4085-4106 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 364 |

Issue number | 8 |

DOIs | |

State | Published - Apr 27 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*364*(8), 4085-4106. https://doi.org/10.1090/S0002-9947-2012-05434-1

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*Transactions of the American Mathematical Society*, vol. 364, no. 8, pp. 4085-4106. https://doi.org/10.1090/S0002-9947-2012-05434-1

**Formulas for the multiplicity of graded algebras.** / Xie, Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Formulas for the multiplicity of graded algebras

AU - Xie, Yu

PY - 2012/4/27

Y1 - 2012/4/27

N2 - 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 ⊆ 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 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 Pl̈ucker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

AB - 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 ⊆ 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 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 Pl̈ucker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

UR - http://www.scopus.com/inward/record.url?scp=84860211883&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860211883&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2012-05434-1

DO - 10.1090/S0002-9947-2012-05434-1

M3 - Article

VL - 364

SP - 4085

EP - 4106

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -