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 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, can be extended to the length of the chain of prime ideals in A.

Lying over and incomparability[edit]

First, we fix some terminology. If \mathfrak{p} and \mathfrak{q} are prime ideals of A and B, respectively, such that

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

(note that \mathfrak{q} \cap A is automatically a prime ideal of A) then we say that \mathfrak{p} lies under \mathfrak{q} and that \mathfrak{q} lies over \mathfrak{p}. 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

\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 ≤ im, \mathfrak{q}_i lies over \mathfrak{p}_i, then the latter chain can be extended to a chain

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

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

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

\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 ≤ im, \mathfrak{q}_i lies over \mathfrak{p}_i, then the latter chain can be extended to a chain

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

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

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.