An <ideal of definition> of a <Noetherian local ring> $(A,\mathfrak m)$ is an $\mathfrak m$-primary ideal $I$; equivalently, $\sqrt I=\mathfrak m$, or some power $\mathfrak m^r$ is contained in $I$.

If $M$ is a finitely generated module of dimension $d$, the <Hilbert–Samuel function>
$$
\chi(M,I;n)=\ell(M/I^nM)
$$
agrees for all sufficiently large $n$ with a polynomial of degree $d$. Its leading term is
$$
\frac{e_I(M)}{d!}n^d,
$$
where $e_I(M)$ is the <Hilbert–Samuel multiplicity>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 1 / a / Solution

{c}
{title2=$e_I(M)$}

If $M$ has <Krull dimension> $d$, the coefficient of $n^d$ in its <Hilbert-Samuel polynomial> is $e_I(M)/d!$. The positive integer $e_I(M)$ is the Hilbert–Samuel multiplicity of $M$ with respect to $I$.

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 Incoming links: Hilbert–Samuel multiplicity