Formulas for the multiplicity of graded algebras

Yu Xie

Research output: Contribution to journalArticle

8 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)4085-4106
Number of pages22
JournalTransactions of the American Mathematical Society
Volume364
Issue number8
DOIs
StatePublished - Apr 27 2012

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Graded Algebra
Algebra
Multiplicity
Artinian Ring
Noetherian
Local Ring
Hyperplane
Hypersurface
Intersection Theory
Nonsingularity
Fibers
Express
Inclusion
Fiber
Ring
Generalise
Standards

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 364, No. 8, 27.04.2012, p. 4085-4106.

Research output: Contribution to journalArticle

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