Associated graded ring (${\mathrm{gr}}_I(R)$)

For an ideal $I$, the associated graded ring is

For an increasing filtration of a module $M_n$ compatible with a filtered algebra $A$, set $\mathrm{gr}M={⨁}_n{M}_n/M_{n-1}$. Multiplication on the quotients makes it a graded module over $\mathrm{gr}A$. Its cumulative homogeneous dimensions equal $\dim M_n$ for a finite-dimensional exhaustive nonnegative filtration of a module.

Give $N$ the subspace filtration and $M/N$ the quotient filtration. If the degree-$i$ class of $m\in F_{i}M$ maps to zero, then $m=n+m^{\prime }$ with $n\in N$ and $m^{\prime }\in F_{i-1}M$. Thus $n\in N\cap F_{i}M$ and the class comes from $\mathrm{gr}N$. Inclusion is injective in every degree and the quotient map is surjective, proving the displayed short exact sequence of associated graded modules. Exactness can fail for arbitrarily chosen incompatible filtrations.

For a Noetherian local ring $(A,\mathfrak{m})$, the pole dimension $d({\mathrm{gr}}_{\mathfrak{m}}(A))$ is the order of the pole at $t=1$ ofThe Hilbert-Serre theorem makes this series rational, so the order is defined.

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, $\mathrm{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.

For a Noetherian local ring $(R,\mathfrak{m})$, let $H_R(t)=\mathrm{length}(R/{\mathfrak{m}}^t)$. Its eventual Hilbert-Samuel polynomial has degree $d(R)$, equivalently the pole order at $1$ of the Hilbert series of ${\mathrm{gr}}_{\mathfrak{m}}R$. This is the local length-growth invariant, not the embedding dimension. The prime-chain lower bound for local length proves $\dim R\le d(R)$; the full local dimension theorem gives equality.

A prime chain of length $s$ in a Noetherian local ring forces its maximal-ideal length function to be at least $c{t}^s$ for large $t$. Quotient by the initial prime to reduce to a domain, choose a nonzero $a$ in the next prime, and use the Artin-Rees lemma to obtain $H_R(t)\ge H_R(t-c_0)+H_{R/(a)}(t)$. Induction on chain length and summing about $t/(2c_0)$ terms prove the bound. This yields $\dim R\le d(R)$ without assuming the Krull height theorem.

For an ideal of definition $I$ of a Noetherian local ring and a finitely generated module $M$, the Hilbert–Samuel function isFor sufficiently large $n$, it equals the Hilbert-Samuel polynomial.

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 $I$.