Analytically normal ring: Difference between revisions

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In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field.

Zariski (1950) proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem. Nagata (1958, 1962, Appendix A1, example 7) gave an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.

References

Nagata, Masayoshi (1958), "An example of a normal local ring which is analytically reducible", Mem. Coll. Sci. Univ. Kyoto. Ser. A Math., 31: 83–85, MR 0097395
Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, New York-London: Interscience Publishers, ISBN 978-0470628652
Zariski, Oscar (1948), "Analytical irreducibility of normal varieties", Annals of Mathematics, Second Series, 49 (2): 352–361, doi:10.2307/1969284, JSTOR 1969284, MR 0024158
Zariski, Oscar (1950), "Sur la normalité analytique des variétés normales", Annales de l'Institut Fourier, 2: 161–164, doi:10.5802/aif.27, MR 0045413
Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876

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-In algebra, an '''analytically irreducible ring''' is a [[local ring]] whose [[completion (ring theory)|completion]] is normal, in other words a domain that is integrally closed in its quotient field.
+In [[algebra]], an '''analytically normal ring''' is a [[local ring]] whose [[completion (ring theory)|completion]] is a [[normal ring]], in other words a [[integral domain|domain]] that is [[integral closure|integrally closed]] in its [[quotient field]].
-{{harvtxt|Zariski|1950}} proved that if a local ring of an algebraic variety is normal, then it is analytically normal. {{harvs|txt|last=Nagata|year1=1958|year2=1962|loc2=Appendix A1, example 7}} gave such an example of a normal Noetherian local ring that is analytically reducible and therefore not analytically normal.
+{{harvtxt|Zariski|1950}} proved that if a local ring of an [[algebraic variety]] is normal, then it is analytically normal, which is in some sense a variation of [[Zariski's main theorem]]. {{harvs|txt|last=Nagata|year1=1958|year2=1962|loc2=Appendix A1, example 7}} gave an example of a normal [[Noetherian ring|Noetherian]] local ring that is [[analytically reducible]] and therefore not analytically normal.
 == References ==
-*{{citation|mr=0097395|last=Nagata|first= Masayoshi|title=An example of a normal local ring which is analytically reducible|journal=Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math.|volume= 31|year= 1958|pages= 83–85|url= http://projecteuclid.org/euclid.kjm/1250776950}}
+*{{citation|mr=0097395|last=Nagata|first= Masayoshi|title=An example of a normal local ring which is analytically reducible|journal=Mem. Coll. Sci. Univ. Kyoto. Ser. A Math.|volume= 31|year= 1958|pages= 83–85|url= http://projecteuclid.org/euclid.kjm/1250776950}}
-*{{citation|authorlink=Masayoshi Nagata|last=Nagata|first= Masayoshi|title=Local rings|series= Interscience Tracts in Pure and Applied Mathematics|volume= 13|publisher= Interscience Publishers|place=New York-London |year=1962}}
+*{{citation|authorlink=Masayoshi Nagata|last=Nagata|first= Masayoshi|title=Local rings|series= Interscience Tracts in Pure and Applied Mathematics|volume= 13|publisher= Interscience Publishers|place=New York-London |year=1962|isbn= 978-0470628652}}
-*{{citation| mr=0024158 |last=Zariski|first= Oscar|authorlink=Oscar Zariski|title=Analytical irreducibility of normal varieties|journal=Ann. of Math. (2) |volume=49|year=1948|pages= 352–361|doi=10.2307/1969284}}
+*{{citation| mr=0024158 |last=Zariski|first= Oscar|authorlink=Oscar Zariski|title=Analytical irreducibility of normal varieties|journal=Annals of Mathematics | series = Second Series |volume=49|year=1948|issue=2|pages= 352–361|doi=10.2307/1969284|jstor=1969284}}
-*{{citation|mr=0045413|last=Zariski|first= Oscar|title=Sur la normalité analytique des variétés normales|journal=Ann. Inst. Fourier Grenoble 2 |year=1950|pages= 161–164|url=http://www.numdam.org/item?id=AIF_1950__2__161_0}}
+*{{citation|mr=0045413|last=Zariski|first= Oscar|title=Sur la normalité analytique des variétés normales|journal=Annales de l'Institut Fourier |volume=2 |year=1950|pages= 161–164|doi=10.5802/aif.27|url=http://www.numdam.org/item?id=AIF_1950__2__161_0|doi-access=free}}
-*{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | origyear=1960 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90171-8 | id={{MathSciNet | id = 0389876}} | year=1975}}
+*{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | last2=Samuel | first2=Pierre | author2-link=Pierre Samuel | title=Commutative algebra. Vol. II | origyear=1960 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90171-8 |mr=0389876 | year=1975}}
 [[Category:Commutative algebra]]
+{{commutative-algebra-stub}}