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As
Hilbert–Poincaré series are additive on exact sequences, the multiplicity is additive on exact sequences of modules of the same dimension.
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The following theorem, due to
Christer Lech, gives a priori bounds for multiplicity.
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828:
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813:
81:
The notion of the multiplicity of a module is a generalization of the
447:{\displaystyle F(t)=\sum _{1}^{d}{a_{d-i} \over (1-t)^{d}}+r(t).}
703:{\displaystyle e(I)\leq d!\deg(R)\lambda (R/{\overline {I}}).}
763:
Integral
Closure: Rees Algebras, Multiplicities, Algorithms
96:
The main focus of the theory is to detect and measure a
766:. Springer Science & Business Media. p. 129.
85:. By Serre's intersection formula, it is linked to an
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is a polynomial. By definition, the multiplicity of
702:
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498:are the coefficients of the Hilbert polynomial of
490:
446:
331:
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243:
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186:. This series is a rational function of the form
116:are intimately connected to multiplicity theory.
826:
759:
502:expanded in binomial coefficients. We have
119:
244:{\displaystyle {\frac {P(t)}{(1-t)^{d}}},}
812:
827:
545:{\displaystyle \mathbf {e} (M)=a_{0}.}
128:be a positively graded ring such that
98:singular point of an algebraic variety
332:{\displaystyle \mathbf {e} (M)=P(1).}
786:
71:{\displaystyle \mathbf {e} _{I}(M).}
609:
581:
13:
760:Vasconcelos, Wolmer (2006-03-30).
14:
846:
789:"Note on multiplicity of ideals"
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617:{\displaystyle {\mathfrak {m}}}
589:{\displaystyle {\mathfrak {m}}}
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83:degree of a projective variety
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465:) is a polynomial. Note that
342:The series may be rewritten
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572:is local with maximal ideal
132:is finitely generated as an
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731:Hilbert–Samuel multiplicity
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104:). Because of this aspect,
102:resolution of singularities
10:
851:
721:Dimension theory (algebra)
87:intersection multiplicity
165:be a finitely generated
120:Multiplicity of a module
34:(often a maximal ideal)
22:multiplicity of a module
835:Theorems in ring theory
491:{\displaystyle a_{d-i}}
184:Hilbert–Poincaré series
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736:Hilbert–Kunz function
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624:-primary ideal, then
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16:In abstract algebra,
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276:{\displaystyle P(t)}
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41:
805:1960ArM.....4...63L
793:Arkiv för Matematik
566: —
91:intersection theory
18:multiplicity theory
814:10.1007/BF02591323
741:Normally flat ring
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787:Lech, C. (1960).
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156:Krull dimension
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726:j-multiplicity
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78:
67:
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139:-algebra and
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110:Rees algebras
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23:
20:concerns the
19:
799:(1): 63–86.
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169:-module and
166:
162:
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151:
150:. Note that
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95:
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31:
24:
21:
17:
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154:has finite
747:References
690:¯
671:λ
659:
647:≤
481:−
408:−
393:−
369:∑
220:−
829:Category
715:See also
596:. If an
568:Suppose
148:Artinian
801:Bibcode
89:in the
770:
457:where
254:where
182:) its
161:. Let
27:at an
100:(cf.
29:ideal
768:ISBN
563:Lech
287:is
124:Let
112:and
809:doi
656:deg
600:is
146:is
831::
807:.
795:.
791:.
108:,
93:.
817:.
811::
803::
797:4
776:.
698:.
695:)
687:I
681:/
677:R
674:(
668:)
665:R
662:(
653:!
650:d
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641:I
638:(
635:e
610:m
598:I
582:m
570:R
540:.
535:0
531:a
527:=
524:)
521:M
518:(
514:e
500:M
484:i
478:d
474:a
463:t
461:(
459:r
442:.
439:)
436:t
433:(
430:r
427:+
419:d
415:)
411:t
405:1
402:(
396:i
390:d
386:a
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373:1
365:=
362:)
359:t
356:(
353:F
327:.
324:)
321:1
318:(
315:P
312:=
309:)
306:M
303:(
299:e
285:M
271:)
268:t
265:(
262:P
239:,
231:d
227:)
223:t
217:1
214:(
209:)
206:t
203:(
200:P
180:t
178:(
175:M
171:F
167:R
163:M
159:d
152:R
144:0
141:R
137:0
134:R
130:R
126:R
66:.
63:)
60:M
57:(
52:I
47:e
32:I
25:M
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