Going up and going down

Going up and going down

Let A ⊆ B be an extension of commutative rings.

The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in B, each member of which lies over members of a longer chain of prime ideals in A, to be able to be extended to the length of the chain of prime ideals in A.

Lying over and incomparability

First, we fix some terminology. If \mathfrak{p} and \mathfrak{q} are prime ideals of A and B, respectively, such that

:\mathfrak{q} \cap A = \mathfrak{p}

(note that \mathfrak{q} \cap A is automatically a prime ideal of A) then we say that \mathfrak{p} lies under \mathfrak{q} and that \mathfrak{q} lies over \mathfrak{p}. In general, a ring extension A ⊆ B of commutative rings is said to satisfy the lying over property if every prime ideal \mathfrak{p} of A lies under some prime ideal \mathfrak{q} of B.

The extension A ⊆ B is said to satisfy the incomparability property if whenever \mathfrak{q} and \mathfrak{q}' are distinct primes of B lying over a prime \mathfrak{p} in A, then \mathfrak{q} ⊈ \mathfrak{q}' and \mathfrak{q}' ⊈ \mathfrak{q}.

Going-up

The ring extension A ⊆ B is said to satisfy the going-up property if whenever

:\mathfrak{p}_1 \subseteq \mathfrak{p}_2 \subseteq !!;\cdots\cdots\cdots!!, \subseteq \mathfrak{p}_n

is a chain of prime ideals of A and

:\mathfrak{q}_1 \subseteq \mathfrak{q}_2 \subseteq \cdots \subseteq \mathfrak{q}_m

is a chain of prime ideals of B with m \mathfrak{q}_i lies over \mathfrak{p}_i for 1 ≤ i ≤ m, then the latter chain can be extended to a chain

:\mathfrak{q}_1 \subseteq \mathfrak{q}_2 \subseteq \cdots \subseteq \mathfrak{q}_m \subseteq \cdots \subseteq \mathfrak{q}_n

such that \mathfrak{q}_i lies over \mathfrak{p}_i for each 1 ≤ i ≤ n.

In it is shown that if an extension A ⊆ B satisfies the going-up property, then it also satisfies the lying-over property.

Going-down

The ring extension A ⊆ B is said to satisfy the going-down property if whenever

:\mathfrak{p}_1 \supseteq \mathfrak{p}_2 \supseteq !!;\cdots\cdots\cdots!!, \supseteq \mathfrak{p}_n

is a chain of prime ideals of A and

:\mathfrak{q}_1 \supseteq \mathfrak{q}_2 \supseteq \cdots \supseteq \mathfrak{q}_m

is a chain of prime ideals of B with m \mathfrak{q}_i lies over \mathfrak{p}_i for 1 ≤ i ≤ m, then the latter chain can be extended to a chain

:\mathfrak{q}_1 \supseteq \mathfrak{q}_2 \supseteq \cdots \supseteq \mathfrak{q}_m \supseteq \cdots \supseteq \mathfrak{q}_n

such that \mathfrak{q}_i lies over \mathfrak{p}_i for each 1 ≤ i ≤ n.

There is a generalization of the ring extension case with ring morphisms. Let f : A → B be a (unital) ring homomorphism so that B is a ring extension of f(A). Then f is said to satisfy the going-up property if the going-up property holds for f(A) in B.

Similarly, if B is a ring extension of f(A), then f is said to satisfy the going-down property if the going-down property holds for f(A) in B.

In the case of ordinary ring extensions such as A ⊆ B, the inclusion map is the pertinent map.

Going-up and going-down theorems

The usual statements of going-up and going-down theorems refer to a ring extension A ⊆ B:

1. (Going up) If B is an integral extension of A, then the extension satisfies the going-up property (and hence the lying over property), and the incomparability property.
2. (Going down) If B is an integral extension of A, and B is a domain, and A is integrally closed in its field of fractions, then the extension (in addition to going-up, lying-over and incomparability) satisfies the going-down property.

There is another sufficient condition for the going-down property:

• If A ⊆ B is a flat extension of commutative rings, then the going-down property holds.

Proof: Let p1 ⊆ p2 be prime ideals of A and let q2 be a prime ideal of B such that q2 ∩ A = p2. We wish to prove that there is a prime ideal q1 of B contained in q2 such that q1 ∩ A = p1. Since A ⊆ B is a flat extension of rings, it follows that A**p2 ⊆ B**q2 is a flat extension of rings. In fact, A**p2 ⊆ B**q2 is a faithfully flat extension of rings since the inclusion map A**p2 → B**q2 is a local homomorphism. Therefore, the induced map on spectra Spec(B**q2) → Spec(A**p2) is surjective and there exists a prime ideal of B**q2 that contracts to the prime ideal p1A**p2 of A**p2. The contraction of this prime ideal of B**q2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete. Q.E.D.

References

• Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969,
• Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.

References

1. This follows from a much more general lemma in Bruns-Herzog, Lemma A.9 on page 415.
2. Matsumura, page 33, (5.D), Theorem 4