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A commutative [[Noetherian ring|Noetherian ring]] satisfying the three axioms stated below. It is known that a [[Geometric ring|geometric ring]] possesses several qualitative properties not inherent in arbitrary Noetherian rings. The concept of an excellent ring makes it possible to take the most important properties of geometric rings axiomatically into account.
 
A commutative [[Noetherian ring|Noetherian ring]] satisfying the three axioms stated below. It is known that a [[Geometric ring|geometric ring]] possesses several qualitative properties not inherent in arbitrary Noetherian rings. The concept of an excellent ring makes it possible to take the most important properties of geometric rings axiomatically into account.
  
Axioms of an excellent ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367601.png" />.
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Axioms of an excellent ring $  A $.
  
A1. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367602.png" /> is a universal chain ring. (A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367603.png" /> is said to be a chain ring if for any two prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367604.png" /> of it the lengths of any two saturated chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367605.png" /> of prime ideals are the same. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367606.png" /> is said to be a universal chain ring if any polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367607.png" /> is a chain ring.)
+
A1. The ring $  A $
 +
is a universal chain ring. (A ring $  A $
 +
is said to be a chain ring if for any two prime ideals $  \mathfrak p \neq \mathfrak p  ^  \prime  $
 +
of it the lengths of any two saturated chains $  \mathfrak p = \mathfrak p _ {0} \subset  \mathfrak p _ {1} \subset  \dots \subset  \mathfrak p _ {n} = \mathfrak p  ^  \prime  $
 +
of prime ideals are the same. A ring $  A $
 +
is said to be a universal chain ring if any polynomial ring $  A [ T _ {1} \dots T _ {k} ] $
 +
is a chain ring.)
  
A2. The formal fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367608.png" /> are geometrically regular, i.e. for any prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e0367609.png" /> and any homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676010.png" /> into a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676011.png" />, the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676012.png" /> is regular. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676013.png" /> is the completion of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676014.png" />.
+
A2. The formal fibres of $  A $
 +
are geometrically regular, i.e. for any prime ideal $  \mathfrak p \subset  A $
 +
and any homomorphism from $  A $
 +
into a field $  K $,  
 +
the ring $  \widehat{A}  _ {\mathfrak p }  \otimes _ {A} K $
 +
is regular. Here $  \widehat{A}  _ {\mathfrak p }  $
 +
is the completion of the local ring $  A _ {\mathfrak p }  $.
  
A3. For any integral finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676015.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676016.png" /> there is a non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676017.png" /> such that the ring of fractions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676018.png" />, is regular.
+
A3. For any integral finite $  A $-
 +
algebra $  B $
 +
there is a non-zero element $  b \in B $
 +
such that the ring of fractions, $  B [ b  ^ {-} 1 ] $,  
 +
is regular.
  
 
Excellent rings possess the following properties:
 
Excellent rings possess the following properties:
  
1) For an excellent ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676019.png" />, the set of regular (normal) points of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676020.png" /> is open.
+
1) For an excellent ring $  A $,  
 +
the set of regular (normal) points of the scheme $  \mathop{\rm Spec}  A $
 +
is open.
  
2) If an excellent local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676021.png" /> is reduced (normal or equi-dimensional), then so is the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676022.png" />.
+
2) If an excellent local ring $  A $
 +
is reduced (normal or equi-dimensional), then so is the completion $  \widehat{A}  $.
  
3) The integral closure of an excellent ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676023.png" /> in a finite extension of the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676024.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676025.png" />-algebra.
+
3) The integral closure of an excellent ring $  A $
 +
in a finite extension of the field of fractions of $  A $
 +
is a finite $  A $-
 +
algebra.
  
4) If a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676026.png" /> is excellent, then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676027.png" />-algebra of finite type is also an excellent ring.
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4) If a ring $  A $
 +
is excellent, then any $  A $-
 +
algebra of finite type is also an excellent ring.
  
Two important examples of excellent rings are the complete local rings (or analytic rings) and the Dedekind rings with field of fractions of characteristic zero. Therefore, the class of excellent rings is sufficiently large and contains, in particular, all algebras of finite type over a field or over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676028.png" /> of integers.
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Two important examples of excellent rings are the complete local rings (or analytic rings) and the Dedekind rings with field of fractions of characteristic zero. Therefore, the class of excellent rings is sufficiently large and contains, in particular, all algebras of finite type over a field or over the ring $  \mathbf Z $
 +
of integers.
  
The excellency of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676029.png" /> is closely connected with the possibility of resolution of singularities of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676030.png" /> (cf. [[#References|[1]]] and [[#References|[2]]]).
+
The excellency of a ring $  A $
 +
is closely connected with the possibility of resolution of singularities of the scheme $  \mathop{\rm Spec}  A $(
 +
cf. [[#References|[1]]] and [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonne, "Eléments de géométrie algébrique" ''Publ. Math. IHES'' , '''2''' (1965) {{MR|0199181}} {{ZBL|0135.39701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero. I" ''Ann. of Math.'' , '''79''' : 1 (1964) pp. 109–203 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, J. Dieudonne, "Eléments de géométrie algébrique" ''Publ. Math. IHES'' , '''2''' (1965) {{MR|0199181}} {{ZBL|0135.39701}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero. I" ''Ann. of Math.'' , '''79''' : 1 (1964) pp. 109–203 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A [[Chain ring|chain ring]] is also called a catenarian ring. A sequence of prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676031.png" /> is saturated if there is no prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676032.png" /> and integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676034.png" />, with both inclusions strict. An excellent ring is a universal Japanese ring. An integer domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676035.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676036.png" /> if for every finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676037.png" /> of its field fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676038.png" /> the integral closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676039.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676041.png" /> is a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676042.png" />-module. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676043.png" /> is universal Japanese if it is Noetherian and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676044.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676045.png" /> for every prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036760/e03676047.png" />. Other terminology for universal Japanese: Nagata ring, pseudo-geometric ring. Cf. also [[Geometric ring|Geometric ring]].
+
A [[Chain ring|chain ring]] is also called a catenarian ring. A sequence of prime ideals $  \mathfrak p _ {0} \subset  \dots \subset  \mathfrak p _ {n} $
 +
is saturated if there is no prime ideal $  \mathfrak q $
 +
and integer $  i \in \{ 0 \dots n - 1 \} $
 +
such that $  \mathfrak p _ {i} \subset  \mathfrak q \subset  \mathfrak p _ {i+} 1 $,  
 +
with both inclusions strict. An excellent ring is a universal Japanese ring. An integer domain $  A $
 +
is $  N _ {2} $
 +
if for every finite extension $  L / K $
 +
of its field fractions $  K $
 +
the integral closure $  A _ {L} $
 +
of $  A $
 +
in $  L $
 +
is a finite $  A $-
 +
module. A ring $  B $
 +
is universal Japanese if it is Noetherian and if $  B / \mathfrak p $
 +
is $  N _ {2} $
 +
for every prime ideal $  \mathfrak p $
 +
of $  B $.  
 +
Other terminology for universal Japanese: Nagata ring, pseudo-geometric ring. Cf. also [[Geometric ring|Geometric ring]].

Latest revision as of 19:38, 5 June 2020


A commutative Noetherian ring satisfying the three axioms stated below. It is known that a geometric ring possesses several qualitative properties not inherent in arbitrary Noetherian rings. The concept of an excellent ring makes it possible to take the most important properties of geometric rings axiomatically into account.

Axioms of an excellent ring $ A $.

A1. The ring $ A $ is a universal chain ring. (A ring $ A $ is said to be a chain ring if for any two prime ideals $ \mathfrak p \neq \mathfrak p ^ \prime $ of it the lengths of any two saturated chains $ \mathfrak p = \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \mathfrak p _ {n} = \mathfrak p ^ \prime $ of prime ideals are the same. A ring $ A $ is said to be a universal chain ring if any polynomial ring $ A [ T _ {1} \dots T _ {k} ] $ is a chain ring.)

A2. The formal fibres of $ A $ are geometrically regular, i.e. for any prime ideal $ \mathfrak p \subset A $ and any homomorphism from $ A $ into a field $ K $, the ring $ \widehat{A} _ {\mathfrak p } \otimes _ {A} K $ is regular. Here $ \widehat{A} _ {\mathfrak p } $ is the completion of the local ring $ A _ {\mathfrak p } $.

A3. For any integral finite $ A $- algebra $ B $ there is a non-zero element $ b \in B $ such that the ring of fractions, $ B [ b ^ {-} 1 ] $, is regular.

Excellent rings possess the following properties:

1) For an excellent ring $ A $, the set of regular (normal) points of the scheme $ \mathop{\rm Spec} A $ is open.

2) If an excellent local ring $ A $ is reduced (normal or equi-dimensional), then so is the completion $ \widehat{A} $.

3) The integral closure of an excellent ring $ A $ in a finite extension of the field of fractions of $ A $ is a finite $ A $- algebra.

4) If a ring $ A $ is excellent, then any $ A $- algebra of finite type is also an excellent ring.

Two important examples of excellent rings are the complete local rings (or analytic rings) and the Dedekind rings with field of fractions of characteristic zero. Therefore, the class of excellent rings is sufficiently large and contains, in particular, all algebras of finite type over a field or over the ring $ \mathbf Z $ of integers.

The excellency of a ring $ A $ is closely connected with the possibility of resolution of singularities of the scheme $ \mathop{\rm Spec} A $( cf. [1] and [2]).

References

[1] A. Grothendieck, J. Dieudonne, "Eléments de géométrie algébrique" Publ. Math. IHES , 2 (1965) MR0199181 Zbl 0135.39701
[2] H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero. I" Ann. of Math. , 79 : 1 (1964) pp. 109–203 MR0199184 Zbl 0122.38603

Comments

A chain ring is also called a catenarian ring. A sequence of prime ideals $ \mathfrak p _ {0} \subset \dots \subset \mathfrak p _ {n} $ is saturated if there is no prime ideal $ \mathfrak q $ and integer $ i \in \{ 0 \dots n - 1 \} $ such that $ \mathfrak p _ {i} \subset \mathfrak q \subset \mathfrak p _ {i+} 1 $, with both inclusions strict. An excellent ring is a universal Japanese ring. An integer domain $ A $ is $ N _ {2} $ if for every finite extension $ L / K $ of its field fractions $ K $ the integral closure $ A _ {L} $ of $ A $ in $ L $ is a finite $ A $- module. A ring $ B $ is universal Japanese if it is Noetherian and if $ B / \mathfrak p $ is $ N _ {2} $ for every prime ideal $ \mathfrak p $ of $ B $. Other terminology for universal Japanese: Nagata ring, pseudo-geometric ring. Cf. also Geometric ring.

How to Cite This Entry:
Excellent ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Excellent_ring&oldid=46867
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article