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[disambiguation needed] 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[edit]

Let AB 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[edit]

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 AB of commutative rings is said to satisfy the lying over property if every prime ideal P of A lies under some prime ideal Q of B.

The extension AB is said to satisfy the incomparability property if whenever Q and Q' are distinct primes of B lying over prime P in A, then QQ' and Q'Q.

Going-up[edit]

The ring extension AB 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 ≤ im, lies over , then the latter chain can be extended to a chain

such that for each 1 ≤ in, lies over .

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

Going down[edit]

The ring extension AB 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 ≤ im, lies over , then the latter chain can be extended to a chain

such that for each 1 ≤ in, lies over .

There is a generalization of the ring extension case with ring morphisms. Let f : AB 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 f(A) is a ring extension, 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 AB, the inclusion map is the pertinent map.

Going-up and going-down theorems[edit]

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

  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 AB is a flat extension of commutative rings, then the going-down property holds.[1]

Proof:[2] Let p1p2 be prime ideals of A and let q2 be a prime ideal of B such that q2A = p2. We wish to prove that there is a prime ideal q1 of B contained in q2 such that q1A = p1. Since AB is a flat extension of rings, it follows that Ap2Bq2 is a flat extension of rings. In fact, Ap2Bq2 is a faithfully flat extension of rings since the inclusion map Ap2Bq2 is a local homomorphism. Therefore, the induced map on spectra Spec(Bq2) → Spec(Ap2) is surjective and there exists a prime ideal of Bq2 that contracts to the prime ideal p1Ap2 of Ap2. The contraction of this prime ideal of Bq2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete. Q.E.D.

References[edit]

  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
  • Atiyah, M. F., and I. G. Macdonald, Introduction to Commutative Algebra, Perseus Books, 1969, ISBN 0-201-00361-9 MR 242802
  • 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
  • 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.