Vipul: 2 revisions

2008-05-12T16:22:58Z

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Vipul: /* Facts */

2008-03-17T00:40:50Z

Facts

← Older revision		Revision as of 00:40, 17 March 2008
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	* The specific Hilbert-Samuel polynomial depends on the specific filtration we choose. However, the ''leading coefficient'' of the Hilbert-Samuel polynomial depends only on the ideal <math>I</math>.		* The specific Hilbert-Samuel polynomial depends on the specific filtration we choose. However, the ''leading coefficient'' of the Hilbert-Samuel polynomial depends only on the ideal <math>I</math>.
	* The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of <math>I</math>.		* The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of <math>I</math>.
			* For a [[Noetherian local ring]] <math>R</math>, the degree of the Hilbert-Samuel polynomial equals the [[Krull dimension]].

Vipul: New page: ==Definition== Let $A$ be a Noetherian local ring with ''unique'' maximal ideal $\mathfrak{m}$ . Let $M$ be a finitely generated $A$ -mod...

2008-03-17T00:34:17Z

New page: ==Definition== Let <math>A</math> be a Noetherian local ring with ''unique'' maximal ideal <math>\mathfrak{m}</math>. Let <math>M</math> be a finitely generated <math>A</math>-mod...

New page

==Definition==

Let <math>A</math> be a [[Noetherian local ring]] with ''unique'' [[maximal ideal]] <math>\mathfrak{m}</math>. Let <math>M</math> be a finitely generated <math>A</math>-module, and <math>I</math> be an [[ideal of finite colength]] in <math>A</math> (in other words, there exists a <math>n</math> such that <math>\mathcal{m}^n \subset I \subset \mathcal{m}</math>).

Consider an essentially <math>I</math>-adic filtration of <math>M</math>, i.e. a descending chain:

<math>M = M_0 \supset M_1 \supset M_2 \supset \ldots</math>

such that there exists <math>n_0</math> such that <math>M_{n+1} = IM_n</math> for <math>n \ge n_0</math>.

The '''Hilbert-Samuel function''' for this filtration is a function sending a positive integer <math>d</math> to the ''length'' of the quotient <math>M_{d-1}/M_d</math> as an <math>A</math>-module. This Hilbert-Samuel function turns out to be equal to a polynomial for sufficiently large values of <math>d</math>. That polynomial is termed the Hilbert-Samuel polynomial for the filtration.

Some people use the term Hilbert-Samuel polynomial for the [[length polynomial]], which is the polynomial measuring the length of the module <math>M_0/M_d</math>.

When we simply talk of the Hilbert-Samuel polynomial, we by default refer to that for the ring as a module over itself. The default ideal is taken to be the maximal ideal and the default filtration is taken to be the standard one.
==Facts==

* The specific Hilbert-Samuel polynomial depends on the specific filtration we choose. However, the ''leading coefficient'' of the Hilbert-Samuel polynomial depends only on the ideal <math>I</math>.
* The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of <math>I</math>.

Hilbert-Samuel polynomial - Revision history

Vipul: 2 revisions

Vipul: /* Facts */

Vipul: New page: ==Definition== Let A be a Noetherian local ring with ''unique'' maximal ideal \mathfrak{m}. Let M be a finitely generated A-mod...

Vipul: New page: ==Definition== Let $A$ be a Noetherian local ring with ''unique'' maximal ideal $\mathfrak{m}$ . Let $M$ be a finitely generated $A$ -mod...