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| In [[commutative algebra]], a branch of [[mathematics]], '''going up''' and '''going down''' are terms which refer to certain properties of [[chain (mathematics)|chain]]s of [[prime ideal]]s in [[integral extension]]s.
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| The phrase '''going up''' refers to the case when a chain can be extended by "upward [[subset|inclusion]]", while '''going down''' refers to the case when a chain can be extended by "downward inclusion".
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| The major results are the '''Cohen–Seidenberg theorems''', which were proved by [[Irvin Cohen|Irvin S. Cohen]] and [[Abraham Seidenberg]]. These are [[colloquialism|colloquially]] known as the '''going-up''' and '''going-down theorems'''.
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| == Going up and going down ==
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| Let ''A''⊆''B'' be an extension of commutative rings.
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| 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''.
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| ===Lying over and incomparability===
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| First, we fix some terminology. If <math>\mathfrak{p}</math> and <math>\mathfrak{q}</math> are [[prime ideal]]s of ''A'' and ''B'', respectively, such that
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| :<math>\mathfrak{q} \cap A = \mathfrak{p}</math>
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| (note that <math>\mathfrak{q} \cap A</math> is automatically a prime ideal of ''A'') then we say that <math>\mathfrak{p}</math> ''lies under'' <math>\mathfrak{q}</math> and that <math>\mathfrak{q}</math> ''lies over'' <math>\mathfrak{p}</math>. In general, a ring extension ''A''⊆''B'' 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''.
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| The extension ''A''⊆''B'' is said to satisfy the '''incomparability property''' if whenever ''Q'' and ''Q' '' are distinct primes of ''B'' lying over prime ''P'' in ''A'', then ''Q''⊈''Q' '' and ''Q' ''⊈''Q''.
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| === Going-up ===
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| The ring extension ''A''⊆''B'' is said to satisfy the '''going-up property''' if whenever
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| :<math>\mathfrak{p}_1 \subseteq \mathfrak{p}_2 \subseteq \cdots \subseteq \mathfrak{p}_n</math>
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| is a chain of [[prime ideal]]s of ''A'' and | |
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| :<math>\mathfrak{q}_1 \subseteq \mathfrak{q}_2 \subseteq \cdots \subseteq \mathfrak{q}_m</math> | |
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| (''m'' < ''n'') is a chain of prime ideals of ''B'' such that for each 1 ≤ ''i'' ≤ ''m'', <math>\mathfrak{q}_i</math> lies over <math>\mathfrak{p}_i</math>, then the latter chain can be extended to a chain
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| :<math>\mathfrak{q}_1 \subseteq \mathfrak{q}_2 \subseteq \cdots \subseteq \mathfrak{q}_n</math>
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| such that for each 1 ≤ ''i'' ≤ ''n'', <math>\mathfrak{q}_i</math> lies over <math>\mathfrak{p}_i</math>.
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| In {{harv|Kaplansky|1970}} it is shown that if an extension ''A''⊆''B'' satisfies the going-up property, then it also satisfies the lying-over property.
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| === Going down ===
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| The ring extension ''A''⊆''B'' is said to satisfy the '''going-down property''' if whenever
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| :<math>\mathfrak{p}_1 \supseteq \mathfrak{p}_2 \supseteq \cdots \supseteq \mathfrak{p}_n</math>
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| is a chain of prime ideals of ''A'' and
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| :<math>\mathfrak{q}_1 \supseteq \mathfrak{q}_2 \supseteq \cdots \supseteq \mathfrak{q}_m</math>
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| (''m'' < ''n'') is a chain of prime ideals of ''B'' such that for each 1 ≤ ''i'' ≤ ''m'', <math>\mathfrak{q}_i</math> lies over <math>\mathfrak{p}_i</math>, then the latter chain can be extended to a chain
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| :<math>\mathfrak{q}_1 \supseteq \mathfrak{q}_2 \supseteq \cdots \supseteq \mathfrak{q}_n</math>
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| such that for each 1 ≤ ''i'' ≤ ''n'', <math>\mathfrak{q}_i</math> lies over <math>\mathfrak{p}_i</math>.
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| 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''.
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| 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''.
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| In the case of ordinary ring extensions such as ''A''⊆''B'', the [[inclusion map]] is the pertinent map.
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| ==Going-up and going-down theorems==
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| The usual statements of going-up and going-down theorems refer to a ring extension ''A''⊆''B'':
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| #(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.
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| #(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.
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| There is another sufficient condition for the going-down property:
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| * If ''A''⊆''B'' is a [[flat extension]] of commutative rings, then the going-down property holds.<ref>This follows from a much more general lemma in Bruns-Herzog, Lemma A.9 on page 415.</ref>
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| ''Proof'':<ref>Matsumura, page 33, (5.D), Theorem 4</ref> Let ''p''<sub>1</sub>⊆''p''<sub>2</sub> be prime ideals of ''A'' and let ''q''<sub>2</sub> be a prime ideal of ''B'' such that ''q''<sub>2</sub> ∩ ''A'' = ''p''<sub>2</sub>. We wish to prove that there is a prime ideal ''q''<sub>1</sub> of ''B'' contained in ''q''<sub>2</sub> such that ''q''<sub>1</sub> ∩ ''A'' = ''p''<sub>1</sub>. Since ''A''⊆''B'' is a flat extension of rings, it follows that ''A''<sub>''p''<sub>2</sub></sub>⊆''B''<sub>''q''<sub>2</sub></sub> is a flat extension of rings. In fact, ''A''<sub>''p''<sub>2</sub></sub>⊆''B''<sub>''q''<sub>2</sub></sub> is a faithfully flat extension of rings since the inclusion map ''A''<sub>''p''<sub>2</sub></sub> → ''B''<sub>''q''<sub>2</sub></sub> is a local homomorphism. Therefore, the induced map on spectra Spec(''B''<sub>''q''<sub>2</sub></sub>) → Spec(''A''<sub>''p''<sub>2</sub></sub>) is surjective and there exists a prime ideal of ''B''<sub>''q''<sub>2</sub></sub> that contracts to the prime ideal ''p''<sub>1</sub>''A''<sub>''p''<sub>2</sub></sub> of ''A''<sub>''p''<sub>2</sub></sub>. The contraction of this prime ideal of ''B''<sub>''q''<sub>2</sub></sub> to ''B'' is a prime ideal ''q''<sub>1</sub> of ''B'' contained in ''q''<sub>2</sub> that contracts to ''p''<sub>1</sub>. The proof is complete. '''Q.E.D.'''
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| ==References==
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| {{Reflist}}
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| * [[Michael Atiyah|Atiyah, M. F.]], and [[I. G. MacDonald]], ''Introduction to Commutative Algebra'', Perseus Books, 1969, ISBN 0-201-00361-9 {{MR|242802}}
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| * 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
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| * [[Irving Kaplansky|Kaplansky, Irving]], ''Commutative rings'', Allyn and Bacon, 1970.
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| * {{cite book|last=Matsumura|first=Hideyuki|title=Commutative algebra|publisher=W. A. Benjamin|year=1970|isbn=978-0-8053-7025-6}}
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| * {{cite book|last=Sharp|first=R. Y.|chapter=13 Integral dependence on subrings (13.38 The going-up theorem, pp. 258–259; 13.41 The going down theorem, pp. 261–262)|title=Steps in commutative algebra|edition=Second|series=London Mathematical Society Student Texts|volume=51|publisher=Cambridge University Press|location=Cambridge|year=2000|pages=xii+355|isbn=0-521-64623-5|mr=1817605}}
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| [[Category:Commutative algebra]]
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| [[Category:Ideals]]
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| [[Category:Prime ideals]]
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| [[Category:Theorems in algebra]]
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I'm Zoila and I live with my husband and our two children in Banksia, in the NSW south part. My hobbies are Vintage car, Juggling and Rugby league football.
Here is my webpage: Fifa 15 coin generator - Full Document,