Going up and going down

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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 Irving S. Cohen and Abraham Seidenberg. These are colloquially known as the going-up and going-down theorems.

1 Going up and going down
- 1.1 Going up
- 1.2 Going down
2 Going-up and going-down properties
3 Reference

[edit] Going up and going down

The going-up and going-down theorems each assert that under certain circumstances, a chain of prime ideals in an integral extension, lying over a longer chain of prime ideals, can be extended to the length of the chain of prime ideals below.

First, we fix some terminology. Let A and B be two commutative rings with unity, and suppose B is an integral extension of A. If $\mathfrak{p}$ and $\mathfrak{q}$ are prime ideals of A and B, respectively, such that

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

then we say that $\mathfrak{p}$ lies under $\mathfrak{q}$ and that $\mathfrak{q}$ lies over $\mathfrak{p}$ .

[edit] Going up

Let A and B be two commutative rings with unity, and suppose B is an integral extension of A. Then whenever

$\mathfrak{p}_1 \subseteq \mathfrak{p}_2 \subseteq \cdots \subseteq \mathfrak{p}_n$

is a chain of prime ideals of A and

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

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, $\mathfrak{q}_i$ lies over $\mathfrak{p}_i$ , then the chain

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

can be extended to a chain

$\mathfrak{q}_1 \subseteq \mathfrak{q}_2 \subseteq \cdots \subseteq \mathfrak{q}_n$

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

[edit] Going down

Let A and B be as above, and furthermore suppose that A is integrally closed. Then whenever

$\mathfrak{p}_1 \supseteq \mathfrak{p}_2 \supseteq \cdots \supseteq \mathfrak{p}_n$

is a chain of prime ideals of A and

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

(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m, $\mathfrak{q}_i$ lies over $\mathfrak{p}_i$ , then the chain

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

can be extended to a chain

$\mathfrak{q}_1 \supseteq \mathfrak{q}_2 \supseteq \cdots \supseteq \mathfrak{q}_n$

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

[edit] Going-up and going-down properties

Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism such that B is an integral extension of f(A). Then f is said to satisfy the going-up property if the conclusion of the going-up theorem holds for f(A) in B.

Similarly, if f(A) is integrally closed, then f is said to satisfy the going-down property if the conclusion of the going-down theorem holds for f(A) in B).

In this context, the usual going-up and going-down theorems simply state that if B is an integral extension of A, then the inclusion map from A to B satisfies the going-up property, and if, in addition, A is integrally closed, then the inclusion map satisfies the going-down property.