# Going up and going down

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 and are prime ideals of *A* and *B*, respectively, such that

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

The extension *A* ⊆ *B* is said to satisfy the **incomparability property** if whenever and are distinct primes of *B* lying over a prime in *A*, then ⊈ and ⊈ .

### Going-up

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

is a chain of prime ideals of *A* and

(*m* < *n*) is a chain of prime ideals of *B* such that for each 1 ≤ *i* ≤ *m*, lies over , then the latter chain can be extended to a chain

such that for each 1 ≤ *i* ≤ *n*, lies over .

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.

### Going-down

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

is a chain of prime ideals of *A* and

(*m* < *n*) is a chain of prime ideals of *B* such that for each 1 ≤ *i* ≤ *m*, lies over , then the latter chain can be extended to a chain

such that for each 1 ≤ *i* ≤ *n*, lies over .

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*:

- (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. - (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 *p*_{1} ⊆ *p*_{2} be prime ideals of *A* and let *q*_{2} be a prime ideal of *B* such that *q*_{2} ∩ *A* = *p*_{2}. We wish to prove that there is a prime ideal *q*_{1} of *B* contained in *q*_{2} such that *q*_{1} ∩ *A* = *p*_{1}. 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 *p*_{1}*A*_{p2} of *A*_{p2}. The contraction of this prime ideal of *B*_{q2} to *B* is a prime ideal *q*_{1} of *B* contained in *q*_{2} that contracts to *p*_{1}. The proof is complete. **Q.E.D.**

## References

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

- 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,
*Commutative rings*, Allyn and Bacon, 1970. - 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.**51**(Second ed.). Cambridge: Cambridge University Press. pp. xii+355. ISBN 0-521-64623-5. MR 1817605.