---
tag: 22J8E7AB8V
kind: definition
name: Hilbert–Samuel function
msc: [13D40, 13H15]
defines: BNQVS83GZV
uses: [MZ02BKPHJA, TMTPWVETQK, CZSCNZH8VY, 05QY1VEBYD, 0670T59HGE]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert–Samuel_function
---

## Definition

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]], let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module, and let $I$ be an [[$\mathfrak m$-primary ideal|TMTPWVETQK]] of $R$.
The *Hilbert–Samuel function* of $M$ with respect to $I$ is the function $\chi_M^I \colon \N \to \N$ given by
$$\chi_M^I(n) = \ell(M/I^n M),$$
where $\ell$ denotes the [[length|CZSCNZH8VY]] of an $R$-module.

## Remarks

The length is finite because $M/I^n M$ is a finitely generated module over the Artinian ring $R/I^n$.
There is a polynomial $P_M^I$ with rational coefficients, the Hilbert–Samuel polynomial, such that $\chi_M^I(n) = P_M^I(n)$ for all large $n$.
If $M \neq 0$, its degree is the Krull dimension of $M$, so it does not depend on $I$.
Example: for $R = k[x, y]_{(x, y)}$, with $k$ a field, $M = R$ and $I = \mathfrak m$, the module $R/\mathfrak m^n$ has the $k$-basis of monomials $x^a y^b$ with $a + b < n$, so $\chi(n) = n(n+1)/2$.
Non-example: for the same ring and $I = (x)$, which is not $\mathfrak m$-primary, the module $R/(x)^n$ has infinite length, so the formula does not define a function to $\N$.
Some authors use $\ell(M/I^{n+1} M)$ instead, which shifts the variable by one.