Hilbert-Samuel polynomial

Definition

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

Consider an essentially $I$-adic filtration of $M$, i.e. a descending chain:

$M=M_{0}\supset M_{1}\supset M_{2}\supset \ldots$

such that there exists $n_{0}$ such that $M_{n+1}=IM_{n}$ for $n\geq n_{0}$.

The Hilbert-Samuel function for this filtration is a function sending a positive integer $d$ to the length of the quotient $M_{d-1}/M_{d}$ as an $A$-module. This Hilbert-Samuel function turns out to be equal to a polynomial for sufficiently large values of $d$. 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 $M_{0}/M_{d}$.

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 $I$.
• The leading coefficient depends on the specific choice of the Hilbert-Samuel polynomial. However, the degree is independent of the choice of $I$.
• For a Noetherian local ring $R$, the degree of the Hilbert-Samuel polynomial equals the Krull dimension.