Multiplicity theory
In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I (often a maximal ideal)
- [math]\displaystyle{ \mathbf{e}_I(M). }[/math]
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 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
- [math]\displaystyle{ \frac{P(t)}{(1-t)^d}, }[/math]
where [math]\displaystyle{ P(t) }[/math] is a polynomial. By definition, the multiplicity of M is
- [math]\displaystyle{ \mathbf{e}(M) = P(1). }[/math]
The series may be rewritten
- [math]\displaystyle{ F(t) = \sum_1^d {a_{d-i} \over (1 - t)^d} + r(t). }[/math]
where r(t) is a polynomial. Note that [math]\displaystyle{ a_{d-i} }[/math] are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We have
- [math]\displaystyle{ \mathbf{e}(M) = a_0. }[/math]
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.[1][2]
Lech — Suppose R is local with maximal ideal [math]\displaystyle{ \mathfrak{m} }[/math]. If an I is [math]\displaystyle{ \mathfrak{m} }[/math]-primary ideal, then
- [math]\displaystyle{ e(I) \le d! \deg(R) \lambda(R/\overline{I}). }[/math]
See also
- Dimension theory (algebra)
- j-multiplicity
- Hilbert–Samuel multiplicity
- Hilbert–Kunz function
- Normally flat ring
References
- ↑ Vasconcelos, Wolmer (2006-03-30) (in en). Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Science & Business Media. pp. 129. ISBN 9783540265030. https://books.google.com/books?id=foG0rKKKXboC.
- ↑ Lech, C. (1960). "Note on multiplicity of ideals". Arkiv för Matematik 4: 63–86. doi:10.1007/BF02591323. http://projecteuclid.org/download/pdf_1/euclid.afm/1485893340.
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