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| In [[mathematics]], a [[commutative ring]] ''R'' is '''catenary''' if for any pair of [[prime ideal]]s
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| :''p'', ''q'',
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| any two strictly increasing chains
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| :''p''=''p''<sub>0</sub> ⊂''p''<sub>1</sub> ... ⊂''p''<sub>''n''</sub>= ''q'' of prime ideals
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| are contained in maximal strictly increasing chains from ''p'' to ''q'' of the same (finite) length. In a geometric situation, in which the [[dimension of an algebraic variety]] attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain ''n'' is usually the difference in dimensions.
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| A ring is called '''universally catenary''' if all finitely generated rings over it are catenary.
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| The word 'catenary' is derived from the Latin word ''catena'', which means "chain".
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| ==Dimension formula==
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| Suppose that ''A'' is a Noetherian domain and ''B'' is a domain containing ''A'' that is finitely generated over ''A''. If ''P'' is a prime ideal of ''B'' and ''p'' its intersection with ''A'', then
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| :<math>\text{height}(P)\le \text{height}(p)+ \text{tr.deg.}_A(B) - \text{tr.deg.}_{\kappa(p)}(\kappa(P)).</math>
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| The '''dimension formula for universally catenary rings''' says that equality holds if ''A'' is universally catenary. Here κ(''P'') is the residue field of ''P'' and tr.deg. means the transcendence degree (of quotient fields). | |
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| ==Examples==
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| Almost all [[Noetherian ring]]s that appear in algebraic geometry are universally catenary.
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| In particular the following rings are universally catenary:
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| *Complete Noetherian [[local ring]]s
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| *[[Dedekind domain]]s (and fields)
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| *[[Cohen-Macaulay ring]]s
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| *Any [[localization of a ring|localization]] of a universally catenary ring
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| *Any finitely generated algebra over a universally catenary ring.
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| ===A ring that is catenary but not universally catenary===
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| It is very hard to construct examples of Noetherian rings that are not universally catenary. The first example was found by {{harvs|txt|first=Masayoshi |last=Nagata|authorlink=Masayoshi Nagata|year1=1956|year2=1962|loc2= page 203 example 2}}, who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
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| Nagata's example is as follows. Choose a field ''k'' and a formal power series ''z''=Σ<sub>''i''>0</sub>''a''<sub>''i''</sub>''x''<sup>''i''</sup> in the ring ''S'' of formal power series in ''x'' over ''k'' such that ''z'' and ''x'' are algebraically independent.
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| Define ''z''<sub>1</sub> = ''z'' and ''z''<sub>''i''+1</sub>=''z''<sub>''i''</sub>/x–''a''<sub>''i''</sub>.
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| Let ''R'' be the (non-Noetherian) ring generated by ''x'' and all the elements ''z''<sub>''i''</sub>.
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| Let ''m'' be the ideal (''x''), and let ''n'' be the ideal generated by ''x''–1 and all the elements ''z''<sub>''i''</sub>. These are both maximal ideals of ''R'', with residue fields isomorphic to ''k''. The local ring ''R''<sub>''m''</sub> is a regular local ring of dimension 1 (the proof of this uses the fact that ''z'' and ''x'' are algebraically independent) and the local ring ''R''<sub>''n''</sub> is a regular Noetherian local ring of dimension 2.
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| Let ''B'' be the localization of ''R'' with respect to all elements not in either ''m'' or ''n''. Then ''B'' is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, ''mB'' (of height 1) and ''nB'' (of height 2).
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| Let ''I'' be the Jacobson radical of ''B'', and let ''A'' = ''k''+''I''. The ring ''A'' is a local domain of dimension 2 with maximal ideal ''I'', so is catenary because all 2-dimensional local domains are catenary. The ring ''A'' is Noetherian because ''B'' is Noetherian and is a finite ''A''-module. However ''A'' is not universally catenary, because if it were then the ideal ''mB'' of ''B'' would have the same height as ''mB''∩''A'' by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(''A'')=2.
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| Nagata's example is also a [[quasi-excellent ring]], so gives an example of a quasi-excellent ring that is not an [[excellent ring]].
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| ==References==
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| *H. Matsumura, ''Commutative algebra'' 1980 ISBN 0-8053-7026-9.
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| *{{citation|mr=0078974|last=Nagata|first= Masayoshi|title=On the chain problem of prime ideals|journal=Nagoya Math. J. |volume=10 |year=1956|pages= 51–64|url=http://projecteuclid.org/euclid.nmj/1118799769}}
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| *Nagata, Masayoshi ''Local rings.'' Interscience Tracts in Pure and Applied Mathematics, No. 13 Interscience Publishers a division of John Wiley & Sons,New York-London 1962, reprinted by R. E. Krieger Pub. Co (1975) ISBN 0-88275-228-6
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| [[Category:Algebraic geometry]]
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| [[Category:Commutative algebra]]
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