Hilbert function: Difference between revisions

Latest revision as of 16:23, 12 May 2008

Definition

Let $R$ be a graded algebra over a field and $M$ a graded module over $R$ . The Hilbert function of $M$ , sometimes denoted $h_{M}$ , is a function that sends any integer $n$ to the dimension of the $n^{t h}$ graded component of $M$ , as a vector space over the underlying field.

We usually consider the Hilbert function for a graded algebra that occurs as a quotient of a multivariate polynomial ring over a field, by a graded ideal. In other words, we study the Hilbert function for a graded algebra over a field that is generated by its degree one terms, and where the degree one component is finite-dimensional as a vector space.

For sufficiently large values, the Hilbert function equals a polynomial, termed the Hilbert polynomial.

Related notions

Hilbert-Samuel function and Hilbert-Samuel polynomial

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 ==Definition==
-Let <math>R</math> be a [[Noetherian ring|Noetherian]] [[local ring]] with maximal ideal <math>M</math> and residue field <math>k = R/M</math>. The '''Hilbert function''' of <math>R</math> is defined as:
+Let <math>R</math> be a [[graded algebra]] over a [[field]] and <math>M</math> a [[graded module]] over <math>R</math>. The '''Hilbert function''' of <math>M</math>, sometimes denoted <math>h_M</math>, is a function that sends any integer <math>n</math> to the dimension of the <math>n^{th}</math> graded component of <math>M</math>, as a vector space over the underlying field.
-<math>H(n) = dim_k (M^n/M^{n+1})</math>
+We usually consider the Hilbert function for a graded algebra that occurs as a quotient of a [[multivariate polynomial ring over a field]], by a [[graded ideal]]. In other words, we study the Hilbert function for a graded algebra over a field that is generated by its degree one terms, and where the degree one component is finite-dimensional as a vector space.
-For large <math>n</math>, <math>H(n)</math> agrees with a polynomial, called the [[Hilbert polynomial]] of <math>R</math>.
+For sufficiently large values, the Hilbert function equals a polynomial, termed the [[Hilbert polynomial]].
+==Related notions==
+* [[Hilbert-Samuel function]] and [[Hilbert-Samuel polynomial]]