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| In [[commutative algebra]] the '''Hilbert–Samuel function''', named after [[David Hilbert]] and [[Pierre Samuel]],<ref>H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.</ref> of a nonzero finitely generated [[module (mathematics)|module]] <math>M</math> over a commutative [[Noetherian]] [[local ring]] <math>A</math> and a [[primary ideal]] <math>I</math> of <math>A</math> is the map <math>\chi_{M}^{I}:\mathbb{N}\rightarrow\mathbb{N}</math> such that, for all <math>n\in\mathbb{N}</math>,
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| :<math>\chi_{M}^{I}(n)=\ell(M/I^{n+1}M)</math>
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| where <math>\ell</math> denotes the [[length of a module|length]] over <math>A</math>. It is related to the [[Hilbert function]] of the [[associated graded module]] <math>\operatorname{gr}_I(M)</math> by the identity
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| : <math>\chi_M^I (n)=\sum_{i=0}^n H(\operatorname{gr}_I(M),i),</math>
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| For sufficiently large <math>n</math>, it coincides with a polynomial function of degree equal to <math>\dim(\operatorname{gr}_I(M))-1</math>.<ref>Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969.</ref>
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| ==Examples==
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| For the [[ring (mathematics)|ring]] of [[formal power series]] in two variables <math>k[[x,y]]</math> taken as a module over itself and graded by the order and the ideal generated by the monomials ''x''<sup>2</sup> and ''y''<sup>3</sup> we have
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| : <math>\chi(1)=1,\quad \chi(2)=3,\quad \chi(3)=5,\quad \chi(4)=6\text{ and } \chi(k)=6\text{ for }k > 4.</math><ref>Ibidem</ref>
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| == Degree bounds ==
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| Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the [[Artin-Rees lemma]]. We denote by <math>P_{I, M}</math> the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
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| Let <math>(R, m)</math> be a Noethrian local ring and ''I'' a m-[[primary ideal]]. If
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| :<math>0 \to M' \to M \to M'' \to 0</math> | |
| is an exact sequence of finitely generated ''R''-modules and if <math>M/I M</math> has finite length,<ref>This implies that <math>M'/IM'</math> and <math>M''/IM''</math> also have finite length.</ref> then we have:<ref>{{harvnb|Eisenbud|loc=Lemma 12.3}}</ref>
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| :<math>P_{I, M} = P_{I, M'} + P_{I, M''} - F</math>
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| where ''F'' is a polynomial of degree strictly less than that of <math>P_{I, M'}</math> and having positive leading coefficient. In particular, if <math>M' \simeq M</math>, then the degree of <math>P_{I, M''}</math> is strictly less than that of <math>P_{I, M} = P_{I, M'}</math>.
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| Proof: Tensoring the given exact sequence with <math>R/I^n</math> and computing the kernel we get the exact sequence:
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| :<math>0 \to (I^n M \cap M')/I^n M' \to M'/I^n M' \to M/I^n M \to M''/I^n M'' \to 0,</math>
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| which gives us:
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| :<math>\chi_M^I(n-1) = \chi_{M'}^I(n-1) + \chi_{M''}^I(n-1) - \ell((I^n M \cap M')/I^n M')</math>.
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| The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large ''n'' and some ''k'',
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| :<math>I^n M \cap M' = I^{n-k} ((I^k M) \cap M') \subset I^{n-k} M'.</math>
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| Thus,
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| :<math>\ell((I^n M \cap M') / I^n M') \le \chi^I_{M'}(n-1) - \chi^I_{M'}(n-k-1)</math>.
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| This gives the desired degree bound.
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| ==References==
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| <references/>
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| * [[David Eisenbud|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
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| {{DEFAULTSORT:Hilbert-Samuel function}}
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| [[Category:Commutative algebra]]
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