Height of a prime

The length of the longest chain of prime ideals contained in a given prime.

Height of a prime

Definition (height of a prime ideal)

Let $R$ be a commutative ring and let $\mathfrak p\in Spec(R)$ .

The height of $\mathfrak p$ , denoted $\operatorname{ht}(\mathfrak p)$ , is

\operatorname{ht}(\mathfrak p) \;:=\; \sup\{\, n \mid \exists\ \mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \cdots \subsetneq \mathfrak p_n=\mathfrak p \text{ in } \operatorname{Spec}(R)\,\}.

\operatorname{ht}(\mathfrak p) \;=\; \dim(R_{\mathfrak p}) \;=\; Krull dimension \text{ of the local ring } R_{\mathfrak p}.

In $\mathbb Z$ .
The prime ideals are $(0)$ and $(p)$ for primes $p$ .
- $\operatorname{ht}((0))=0$ .
- $\operatorname{ht}((p))=1$ , since $(0)\subsetneq(p)$ is a maximal chain.
In $k[x,y]$ .
Let $k$ be a field.
- $\operatorname{ht}((x))=1$ because $(0)\subsetneq(x)$ is a chain and $(x)$ is not maximal.
- $\operatorname{ht}((x,y))=2$ because $(0)\subsetneq(x)\subsetneq(x,y)$ is a chain of length $2$ , and $(x,y)$ is maximal.
Height of a maximal ideal in a polynomial ring.
In $k[x_1,\dots,x_n]$ , the “origin” maximal ideal $(x_1,\dots,x_n)$ has height $n$ (matching $\dim(k[x_1,\dots,x_n])=n$ ).