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Multiplicity theory

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In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I (often a maximal ideal)

e I ( M ) . {\displaystyle \mathbf {e} _{I}(M).}

The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By Serre's intersection formula, it is linked to an intersection multiplicity in the intersection theory.

The main focus of the theory is to detect and measure a singular point of an algebraic variety (cf. resolution of singularities). Because of this aspect, valuation theory, Rees algebras and integral closure are intimately connected to multiplicity theory.

Multiplicity of a module

Let R be a positively graded ring such that R is finitely generated as an R0-algebra and R0 is Artinian. Note that R has finite Krull dimension d. Let M be a finitely generated R-module and FM(t) its Hilbert–Poincaré series. This series is a rational function of the form

P ( t ) ( 1 − t ) d , {\displaystyle {\frac {P(t)}{(1-t)^{d}}},}

where P ( t ) {\displaystyle P(t)} is a polynomial. By definition, the multiplicity of M is

e ( M ) = P ( 1 ) . {\displaystyle \mathbf {e} (M)=P(1).}

The series may be rewritten

F ( t ) = ∑ 1 d a d − i ( 1 − t ) d + r ( t ) . {\displaystyle F(t)=\sum _{1}^{d}{a_{d-i} \over (1-t)^{d}}+r(t).}

where r(t) is a polynomial. Note that a d − i {\displaystyle a_{d-i}} are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We have

e ( M ) = a 0 . {\displaystyle \mathbf {e} (M)=a_{0}.}

As Hilbert–Poincaré series are additive on exact sequences, the multiplicity is additive on exact sequences of modules of the same dimension.

The following theorem, due to Christer Lech, gives a priori bounds for multiplicity.

Lech — Suppose R is local with maximal ideal m {\displaystyle {\mathfrak {m}}} . If an I is m {\displaystyle {\mathfrak {m}}} -primary ideal, then

e ( I ) ≤ d ! deg ⁡ ( R ) λ ( R / I ¯ ) . {\displaystyle e(I)\leq d!\deg(R)\lambda (R/{\overline {I}}).}

See also

References

  1. Vasconcelos, Wolmer (2006-03-30). Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Science & Business Media. p. 129. ISBN 9783540265030.
  2. Lech, C. (1960). "Note on multiplicity of ideals". Arkiv för Matematik. 4 (1): 63–86. Bibcode:1960ArM.....4...63L. doi:10.1007/BF02591323.
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