Height and Chains of Primes

The height of a prime ideal measures its position in the lattice of prime ideals, providing the local counterpart to the global notion of Krull dimension.

Height of Prime Ideals

Definition

Let $R$ be a commutative ring and $\mathfrak{p}$ a prime ideal. The height of $\mathfrak{p}$ , denoted $\operatorname{ht}(\mathfrak{p})$ , is the supremum of lengths $n$ of chains of prime ideals $\mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{p}$ descending from $\mathfrak{p}$ . The coheight of $\mathfrak{p}$ is $\operatorname{coht}(\mathfrak{p}) = \dim(R/\mathfrak{p})$ .

In a Noetherian ring, the height of a prime ideal equals the Krull dimension of the localization: $\operatorname{ht}(\mathfrak{p}) = \dim(R_\mathfrak{p})$ .

Definition

The height of an ideal $I \subseteq R$ is $\operatorname{ht}(I) = \inf\{\operatorname{ht}(\mathfrak{p}) : \mathfrak{p} \supseteq I,\; \mathfrak{p} \text{ prime}\}$ . For a Noetherian ring, this equals the minimum height among the minimal primes over $I$ .

Properties

ExampleHeights in polynomial rings

In $R = k[x_1, \ldots, x_n]$ over a field $k$ :

$\operatorname{ht}((x_1)) = 1$ since $(0) \subsetneq (x_1)$ and $(0)$ is the unique prime below $(x_1)$
$\operatorname{ht}((x_1, x_2)) = 2$ with chain $(0) \subsetneq (x_1) \subsetneq (x_1, x_2)$
$\operatorname{ht}((x_1, \ldots, x_r)) = r$

In general, $\operatorname{ht}(\mathfrak{p}) + \operatorname{coht}(\mathfrak{p}) = n$ for all prime ideals $\mathfrak{p}$ in $k[x_1, \ldots, x_n]$ .

Theorem6.3Height-Coheight Inequality

In a Noetherian ring $R$ that is a finitely generated algebra over a field (or more generally, a catenary ring), every prime ideal $\mathfrak{p}$ satisfies $\operatorname{ht}(\mathfrak{p}) + \dim(R/\mathfrak{p}) \leq \dim(R)$ Equality holds for all primes if and only if $R$ is equidimensional and catenary.

RemarkCatenary rings

A ring is catenary if for every pair of prime ideals $\mathfrak{p} \subseteq \mathfrak{q}$ , all maximal chains of primes between $\mathfrak{p}$ and $\mathfrak{q}$ have the same length. Most rings encountered in algebraic geometry (finitely generated algebras over fields, localizations thereof, complete local rings) are catenary, but non-catenary Noetherian rings do exist (Nagata's examples).