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Difference between revisions of "Geometric ring"

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A [[Local ring|local ring]] of an algebraic variety or a completion of such a ring. A commutative ring obtained from a ring of polynomials over a field by means of the operations of completion, localization and factorization by a prime ideal is called an algebro-geometric ring [[#References|[3]]]. A local ring of an irreducible algebraic variety does not obtain nilpotent elements as a result of completion [[#References|[2]]]. This property of a local ring is known as analytic reducibility. A similar fact concerning local rings of normal varieties [[#References|[1]]] is that the completion of a local ring of a normal algebraic variety is a normal ring (analytic normality). Examples of local Noetherian rings that are not analytically reduced or analytically normal are known [[#References|[4]]]. A pseudo-geometric ring is a [[Noetherian ring|Noetherian ring]] any quotient ring of which by a prime ideal is a Japanese ring. An integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044300/g0443001.png" /> is called a Japanese ring if its integral closure in a finite extension of the field of fractions is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044300/g0443002.png" />-module [[#References|[5]]]. The class of pseudo-geometric rings is closed with respect to localizations and extensions of finite type; it includes the ring of integers and all complete local rings. See also [[Excellent ring|Excellent ring]].
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A [[Local ring|local ring]] of an algebraic variety or a completion of such a ring. A commutative ring obtained from a ring of polynomials over a field by means of the operations of completion, localization and factorization by a prime ideal is called an algebro-geometric ring [[#References|[3]]]. A local ring of an irreducible algebraic variety does not obtain nilpotent elements as a result of completion [[#References|[2]]]. This property of a local ring is known as analytic reducibility. A similar fact concerning local rings of normal varieties [[#References|[1]]] is that the completion of a local ring of a normal algebraic variety is a normal ring (analytic normality). Examples of local Noetherian rings that are not analytically reduced or analytically normal are known [[#References|[4]]]. A pseudo-geometric ring is a [[Noetherian ring|Noetherian ring]] any quotient ring of which by a prime ideal is a Japanese ring. An integral domain $A$ is called a Japanese ring if its integral closure in a finite extension of the field of fractions is a finite $A$-module [[#References|[5]]]. The class of pseudo-geometric rings is closed with respect to localizations and extensions of finite type; it includes the ring of integers and all complete local rings. See also [[Excellent ring|Excellent ring]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,  "Intersection of algebraic and algebroid varieties"  ''Trans. Amer. Math. Soc.'' , '''57'''  (1945)  pp. 1–85</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Samuel,  "Algèbre locale" , Gauthier-Villars  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Chevalley,  "Intersection of algebraic and algebroid varieties"  ''Trans. Amer. Math. Soc.'' , '''57'''  (1945)  pp. 1–85</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Samuel,  "Algèbre locale" , Gauthier-Villars  (1953)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR></table>

Revision as of 09:27, 27 April 2014

A local ring of an algebraic variety or a completion of such a ring. A commutative ring obtained from a ring of polynomials over a field by means of the operations of completion, localization and factorization by a prime ideal is called an algebro-geometric ring [3]. A local ring of an irreducible algebraic variety does not obtain nilpotent elements as a result of completion [2]. This property of a local ring is known as analytic reducibility. A similar fact concerning local rings of normal varieties [1] is that the completion of a local ring of a normal algebraic variety is a normal ring (analytic normality). Examples of local Noetherian rings that are not analytically reduced or analytically normal are known [4]. A pseudo-geometric ring is a Noetherian ring any quotient ring of which by a prime ideal is a Japanese ring. An integral domain $A$ is called a Japanese ring if its integral closure in a finite extension of the field of fractions is a finite $A$-module [5]. The class of pseudo-geometric rings is closed with respect to localizations and extensions of finite type; it includes the ring of integers and all complete local rings. See also Excellent ring.

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[2] C. Chevalley, "Intersection of algebraic and algebroid varieties" Trans. Amer. Math. Soc. , 57 (1945) pp. 1–85
[3] P. Samuel, "Algèbre locale" , Gauthier-Villars (1953)
[4] M. Nagata, "Local rings" , Interscience (1962)
[5] A. Grothendieck, "Eléments de géometrie algébrique IV. Etude locale des schémas et des morphismes des schémas" Publ. Math. IHES : 32 (1967)
How to Cite This Entry:
Geometric ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometric_ring&oldid=16165
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article