Multiplicity theory: Difference between revisions
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It is, by definition, the multiplicity of ''M''. It is additive on exact sequences since Hilbert–Poincaré series are additive. |
It is, by definition, the multiplicity of ''M''. It is additive on exact sequences since Hilbert–Poincaré series are additive. |
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The following theorem, due to Christer Lech, gives a priori bounds for multiplicity.<ref>{{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|year=|isbn=9783540265030|location=|pages=129|language=en}}</ref><ref>{{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|pages=63–86|doi=10.1007/BF02591323|via=}}</ref> |
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The following theorem gives a priori bounds for multiplicity. |
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{{math_theorem |
{{math_theorem |
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|Suppose ''R'' is local with maximal ideal <math>\mathfrak{m}</math>. If an ''I'' is <math>\mathfrak{m}</math>-primary ideal, then |
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:<math>e(I) \le d! \operatorname{deg}(R) \lambda(R/\overline{I}).</math>| name = Lech |
:<math>e(I) \le d! \operatorname{deg}(R) \lambda(R/\overline{I}).</math>| name = Lech |
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Revision as of 01:00, 26 August 2017
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In abstract algebra, multiplicity theory concerns the multiplicity of a module M at an ideal I (often a maximal ideal)
- .
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. Since FM(t) is a rational function, it can be written as
where r(t) is a polynomial. Note that are the coefficients of the Hilbert polynomial of M expanded in binomial coefficients. We let
- .
It is, by definition, the multiplicity of M. It is additive on exact sequences since Hilbert–Poincaré series are additive.
The following theorem, due to Christer Lech, gives a priori bounds for multiplicity.[1][2]
Lech — Suppose R is local with maximal ideal . If an I is -primary ideal, then
See also
References
- ^ Vasconcelos, Wolmer (2006-03-30). Integral Closure: Rees Algebras, Multiplicities, Algorithms. Springer Science & Business Media. p. 129. ISBN 9783540265030.
- ^ Lech, C. (1960). "Note on multiplicity of ideals". Arkiv för Matematik. 4: 63–86. doi:10.1007/BF02591323.