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multiplicity theory#multiplicity of a module

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

:\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

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

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

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

The series may be rewritten

: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} are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We have

:\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.{{Cite book|url=https://books.google.com/books?id=foG0rKKKXboC|title=Integral Closure: Rees Algebras, Multiplicities, Algorithms|last=Vasconcelos|first=Wolmer|date=2006-03-30|publisher=Springer Science & Business Media|isbn=9783540265030|pages=129|language=en}}{{Cite journal|last=Lech|first=C.|date=1960|title=Note on multiplicity of ideals|url=http://projecteuclid.org/download/pdf_1/euclid.afm/1485893340|journal=Arkiv för Matematik|volume=4|issue=1 |pages=63–86|doi=10.1007/BF02591323|bibcode=1960ArM.....4...63L |doi-access=free|url-access=subscription}}

{{math_theorem

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

:e(I) \le d! \deg(R) \lambda(R/\overline{I}).| name = Lech

}}

See also

References

{{reflist}}

Category:Theorems in ring theory