Going up and going down

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

Contents

Going up and going down

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

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}

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

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 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 ≤ 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 satsifies the lying-over property.

Going down

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 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 ≤ 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

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

  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 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
  • Kaplansky, Irving, Commutative rings, Allyn and Bacon, 1970.
  • Matsumura, Hideyuki (1970). Commutative algebra. W. A. Benjamin. ISBN 978-0805370256. 
  • 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. MR1817605. 

Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Going Down — may refer to: * Going Down (novel), a novel * Going Down (song), written by Don Nix and originally performed by Freddie King, a mainstay of blues and rock musicians * Going up and going down, a mathematical concept in commutative algebra * Going… …   Wikipedia

  • Going Down — Going Down …   Википедия

  • Going Down for the Third Time — is a song written and composed by Holland Dozier Holland and recorded by Motown singing group The Supremes in 1967. The song was issued as the b side to the popular Reflections .ong informationThe song features a fierce sounding Diana Ross… …   Wikipedia

  • Going Down (album) — Infobox Album Name = Going Down Type = Remix Album Artist = Client Released = 2004 Recorded = Genre = Electroclash Length = mm:ss Label = Toast Hawaii Producer = Reviews = Last album = Client (2003) This album = Going Down (2004) Next album =… …   Wikipedia

  • As I Was Going Down Sackville Street — As I Was Going Down Sackville Street: A Phantasy in Fact is a book by Oliver St. John Gogarty. Published in 1937, it was Gogarty s first extended prose work and was described by its author as something new in form: neither a memoir nor a novel… …   Wikipedia

  • I'm Going Down (Rose Royce song) — Single infobox Name = I m Going Down from Album = Released = March 31, 1977 Artist = Rose Royce Chart Position = * #27 U.S. * 1 U.S. R B Label = Recorded = 1976 Last single = Do Your Dance (Part 1) (1977) This single = I m Going Down (1977) Next… …   Wikipedia

  • Something Heavy Going Down — Infobox Album | Name = Something Heavy Going Down Type = Live album Artist = Golden Earring Released = 1984 Recorded = ??? Genre = Hard rock Length = 63:51 Label = Twenty One Producer = Golden Earring Shell Schellekens Reviews = *Allmusic… …   Wikipedia

  • Down (band) — Down Down live in 2008 Background information Origin New Orleans, Louisiana, USA Genres …   Wikipedia

  • Down GAA — Irish: An Dún Province: Ulster Nickname(s): The Mournemen (football) The Ardsmen (hurling) …   Wikipedia

  • Down (Jay Sean song) — Down Single by Jay Sean featuring Lil Wayne from the album All or Nothing …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”