Knowlpedia

Height of a prime

The codimension of a prime ideal, measured by the maximum length of chains of primes ending at it.

Let RR be a and let p\mathfrak p be a prime ideal (i.e., p∈Spec⁡(R)\mathfrak p\in\operatorname{Spec}(R), the ).

The height of p\mathfrak p, denoted ht⁡(p)\operatorname{ht}(\mathfrak p), is the supremum of integers n≥0n\ge 0 such that there exists a strictly increasing chain of prime ideals

p0⊊p1⊊⋯⊊pn=p\mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \cdots \subsetneq \mathfrak p_n=\mathfrak p

in RR.

Equivalent characterizations

Equivalently,

ht⁡(p)=dim⁡(Rp),\operatorname{ht}(\mathfrak p)=\dim(R_{\mathfrak p}),

where RpR_{\mathfrak p} is the and dim⁡\dim denotes . In particular, the dimension of RR is the supremum of the heights of its prime ideals.

Examples

  1. The ring of integers. In R=ZR=\mathbb{Z}, one has ht⁡((0))=0\operatorname{ht}((0))=0, and for any prime number pp,
    (0)⊊(p)(0)\subsetneq (p)
    is a maximal chain, so ht⁡((p))=1\operatorname{ht}((p))=1.
  1. A polynomial ring in two variables. In k[x,y]k[x,y] (for a field kk), the prime ideals satisfy
    (0)⊊(x)⊊(x,y),(0)\subsetneq (x)\subsetneq (x,y),
    so ht⁡((0))=0\operatorname{ht}((0))=0, ht⁡((x))=1\operatorname{ht}((x))=1, and ht⁡((x,y))=2\operatorname{ht}((x,y))=2.
  1. Dedekind domains. If RR is a (for instance, the ring of integers in a number field), then every nonzero prime ideal has height 11. In a , the unique nonzero prime ideal also has height 11, reflecting that these rings are “one-dimensional.”