Going up and going down explained

In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions.

The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion".

The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems.

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

ak{p} and ak{q} are prime ideals of A and B, respectively, such thatak{q} \cap A = ak{p}

(note that

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

The extension A ⊆ B is said to satisfy the incomparability property if whenever

ak{q} and ak{q}' are distinct primes of B lying over a prime ak{p} in A, then ak{q}  ⊈ ak{q}' and ak{q}'  ⊈ ak{q} .

Going-up

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

ak{p} _{1} \subseteq ak{p} _{2} \subseteq … … … \subseteq ak{p} _{n}

is a chain of prime ideals of A and

ak{q} _{1} \subseteq ak{q} _{2} \subseteq … \subseteq ak{q} _{m}

is a chain of prime ideals of B with m < n and such that

ak{q} _{i} lies over ak{p} _{i} for 1 ≤ i ≤ m, then the latter chain can be extended to a chainak{q} _{1} \subseteq ak{q} _{2} \subseteq … \subseteq ak{q} _{m} \subseteq … \subseteq ak{q} _{n}

such that

ak{q} _{i} lies over ak{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

ak{p} _{1} \supseteq ak{p} _{2} \supseteq … … … \supseteq ak{p} _{n}

is a chain of prime ideals of A and

ak{q} _{1} \supseteq ak{q} _{2} \supseteq … \supseteq ak{q} _{m}

is a chain of prime ideals of B with m < n and such that

ak{q} _{i} lies over ak{p} _{i} for 1 ≤ i ≤ m, then the latter chain can be extended to a chainak{q} _{1} \supseteq ak{q} _{2} \supseteq … \supseteq ak{q} _{m} \supseteq … \supseteq ak{q} _{n}

such that

ak{q} _{i} lies over ak{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.^[1]

Proof:^[2] 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.
• I.S.. Cohen. A.. Seidenberg. Prime ideals and integral dependence . 1946 . Bull. Amer. Math. Soc. . 52 . 4 . 252–261 . 0015379 . 10.1090/s0002-9904-1946-08552-3. free.
• Book: Kaplansky, Irving . Irving Kaplansky . Commutative rings . Allyn and Bacon . 1970.
• Book: Matsumura, Hideyuki. Commutative algebra. W. A. Benjamin. 1970. 978-0-8053-7025-6.
• Book: Sharp, R. Y.. 13 Integral dependence on subrings (13.38 The going-up theorem, pp. 258–259; 13.41 The going down theorem, pp. 261–262). Steps in commutative algebra. Second. London Mathematical Society Student Texts. 51. Cambridge University Press. Cambridge. 2000. xii+355. 0-521-64623-5. 1817605.

Notes and 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