Height of an ideal

The minimum of the heights of the prime ideals containing the ideal. The height $ \mathop{\rm ht} ( \mathfrak p ) $ of a prime ideal $ \mathfrak p $ in a ring $ A $ is the largest number $ h $( or $ \infty $ if such a number does not exist) such that there exists a chain of different prime ideals

$$ \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \ \mathfrak p _ {h} = \mathfrak p . $$

The co-height $ \mathop{\rm coht} ( \mathfrak p ) $ of a prime ideal $ \mathfrak p $ is defined as the largest $ h $ for which there exists a chain of prime ideals

$$ \mathfrak p = \ \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \mathfrak p _ {h} \neq A. $$

In other words,

$$ \mathop{\rm ht} ( \mathfrak p ) = \mathop{\rm dim} ( A _ {\mathfrak p } ),\ \ \mathop{\rm coht} ( \mathfrak p ) = \mathop{\rm dim} ( A / \mathfrak p ), $$

where dim denotes the dimension of the corresponding Krull ring. The height of a prime ideal is equal to the codimension of the variety defined by the ideal, while the co-height equals the dimension of this variety. The height and the co-height of a prime ideal are connected by the inequality

$$ \mathop{\rm ht} ( \mathfrak p ) + \mathop{\rm coht} ( \mathfrak p ) \leq \mathop{\rm dim} A, $$

which becomes an equality if, for example, $ A $ is a local Cohen–Macaulay ring.

The prime ideals of height zero are the minimal prime ideals. The existence of prime ideals of height one in Noetherian integral domains is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. Krull ring). A more general result — Krull's theorem — interconnects the height with the number of generators of the ideal: In a Noetherian ring the height of an ideal generated by $ r $ elements is not larger than $ r $, and conversely: A prime ideal of height $ r $ is the smallest of all prime ideals containing some $ r $ elements. In particular, any ideal in a Noetherian ring has finite height; this is not true of the co-height [2].

References

[1]	W. Krull, "Primidealketten in allgemeinen Ringbereichen" , Berlin-Leipzig (1928)
[2]	M. Nagata, "Local rings" , Interscience (1962)
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)