Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module ${\displaystyle M}$ over a commutative Noetherian local ring ${\displaystyle A}$ and a primary ideal ${\displaystyle I}$ of ${\displaystyle A}$ is the map ${\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }$ such that, for all ${\displaystyle n\in \mathbb {N} }$,

${\displaystyle \chi _{M}^{I}(n)=\ell (M/I^{n}M)}$

where ${\displaystyle \ell }$ denotes the length over ${\displaystyle A}$. It is related to the Hilbert function of the associated graded module ${\displaystyle \operatorname {gr} _{I}(M)}$ by the identity

${\displaystyle \chi _{M}^{I}(n)=\sum _{i=0}^{n}H(\operatorname {gr} _{I}(M),i).}$

For sufficiently large ${\displaystyle n}$, it coincides with a polynomial function of degree equal to ${\displaystyle \dim(\operatorname {gr} _{I}(M))}$, often called the Hilbert-Samuel polynomial (or Hilbert polynomial).