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Going up and going down

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

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.

First, we fix some terminology. If ${\displaystyle {\mathfrak {p}}}$ and ${\displaystyle {\mathfrak {q}}}$ are prime idealsofA and B, respectively, such that

${\displaystyle {\mathfrak {q}}\cap A={\mathfrak {p}}}$

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

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

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

${\displaystyle {\mathfrak {p}}_{1}\subseteq {\mathfrak {p}}_{2}\subseteq \!\!\;\cdots \cdots \cdots \!\!\,\subseteq {\mathfrak {p}}_{n}}$

is a chain of prime ideals of A and

${\displaystyle {\mathfrak {q}}_{1}\subseteq {\mathfrak {q}}_{2}\subseteq \cdots \subseteq {\mathfrak {q}}_{m}}$

is a chain of prime ideals of B with m < n and such that ${\displaystyle {\mathfrak {q}}_{i}}$ lies over ${\displaystyle {\mathfrak {p}}_{i}}$ for 1 ≤ i ≤ m, then the latter chain can be extended to a chain

${\displaystyle {\mathfrak {q}}_{1}\subseteq {\mathfrak {q}}_{2}\subseteq \cdots \subseteq {\mathfrak {q}}_{m}\subseteq \cdots \subseteq {\mathfrak {q}}_{n}}$

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

In (Kaplansky 1970) it is shown that if an extension A ⊆ B satisfies the going-up property, then it also satisfies the lying-over property.

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

${\displaystyle {\mathfrak {p}}_{1}\supseteq {\mathfrak {p}}_{2}\supseteq \!\!\;\cdots \cdots \cdots \!\!\,\supseteq {\mathfrak {p}}_{n}}$

is a chain of prime ideals of A and

${\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{m}}$

is a chain of prime ideals of B with m < n and such that ${\displaystyle {\mathfrak {q}}_{i}}$ lies over ${\displaystyle {\mathfrak {p}}_{i}}$ for 1 ≤ i ≤ m, then the latter chain can be extended to a chain

${\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{m}\supseteq \cdots \supseteq {\mathfrak {q}}_{n}}$

such that ${\displaystyle {\mathfrak {q}}_{i}}$ lies over ${\displaystyle {\mathfrak {p}}_{i}}$ 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.

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 extensionofA, 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:

• IfA ⊆ 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 q₁ofB 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_p2ofA_p2. The contraction of this prime ideal of B_q2toB is a prime ideal q₁ofB contained in q₂ that contracts to p₁. The proof is complete. Q.E.D.

• Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9 MR242802
• Winfried Bruns; Jürgen Herzog, Cohen–Macaulay rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1
• Cohen, I.S.; Seidenberg, A. (1946). "Prime ideals and integral dependence". Bull. Amer. Math. Soc. 52 (4): 252–261. doi:10.1090/s0002-9904-1946-08552-3. MR 0015379.
• Kaplansky, Irving (1970). Commutative rings. Allyn and Bacon.
• Matsumura, Hideyuki (1970). Commutative algebra. W. A. Benjamin. ISBN 978-0-8053-7025-6.
• Sharp, R. Y. (2000). "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. London Mathematical Society Student Texts. Vol. 51 (Second ed.). Cambridge: Cambridge University Press. pp. xii+355. ISBN 0-521-64623-5. MR 1817605.