---
tag: KW3HY77V6D
kind: statement
role: theorem
name: Dimension theorem for local rings
msc: [13C15, 13H15, 13D40]
uses: [MZ02BKPHJA, MVGBS3J1P7, TMTPWVETQK, BNQVS83GZV, CZSCNZH8VY, 0670T59HGE]
about: [MVGBS3J1P7, MZ02BKPHJA]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_theory_(algebra)
---

## Statement

Let $(R, \mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].
For $n \ge 0$ let $\chi(n) = \ell(R/\mathfrak m^n)$ be the [[length|CZSCNZH8VY]] of $R/\mathfrak m^n$; this is the [[Hilbert–Samuel function|BNQVS83GZV]] of $R$.
Then $\chi(n)$ agrees with a polynomial in $n$ for all large $n$, and the following three integers are equal:

1. the [[Krull dimension|MVGBS3J1P7]] $\dim R$;
2. the least number $\delta(R)$ of elements of $\mathfrak m$ that generate an [[$\mathfrak m$-primary ideal|TMTPWVETQK]];
3. the degree $d(R)$ of the polynomial that agrees with $\chi(n)$ for large $n$.

## Remarks

The usual proof shows $\delta(R) \ge d(R) \ge \dim R \ge \delta(R)$.
The first inequality holds because the Hilbert–Samuel polynomial of an ideal generated by $s$ elements has degree at most $s$, and its degree does not depend on the $\mathfrak m$-primary ideal.
The second goes by induction on $d(R)$.
Take a chain of primes $\mathfrak p_0 \subsetneq \cdots \subsetneq \mathfrak p_r$ and an element $x \in \mathfrak p_1 \setminus \mathfrak p_0$.
Then the ring $R/(\mathfrak p_0 + (x))$ has a smaller $d$ and still contains a chain of length $r - 1$.
The third goes by building, with the [[prime avoidance lemma|FMP5AWG6PR]], elements $x_1, \dots, x_{\dim R}$ such that each $x_i$ lies outside every prime of height $i - 1$ that contains $x_1, \dots, x_{i-1}$.
For example, for $R = k[x, y]_{(x, y)}$ with $k$ a field, $\chi(n) = n(n+1)/2$, of degree $2$, the ideal $\mathfrak m = (x, y)$ needs two generators, and $\dim R = 2$.
Consequences: $\dim R \le \dim_k \mathfrak m/\mathfrak m^2$ with $k = R/\mathfrak m$, so $\dim R$ is finite, and [[Krull's height theorem|TBRTZMHFTV]] follows.

## Explanations

### L1 human

There are three natural ways to measure the size of a local ring.
One counts the longest chain of prime ideals.
One counts how many equations are needed to cut out the point.
One measures how fast the ring grows modulo higher and higher powers of the maximal ideal.
The theorem says that the three counts always give the same number.

### L2 human

The theorem turns the order-theoretic Krull dimension into something computable.
The number $\delta(R)$ gives upper bounds, since any $\mathfrak m$-primary ideal with $s$ generators shows $\dim R \le s$.
The degree $d(R)$ connects dimension with [[Hilbert polynomials|RMH4FBCFDA]] and with the [[multiplicity|JX20MNDDXT]] of the ring.
A list of $\dim R$ elements that generates an $\mathfrak m$-primary ideal is a [[system of parameters|95QTEPMDA4]].