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A transition from a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602801.png" /> to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602803.png" /> is a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602804.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602805.png" /> can be defined as the solution of the problem of a universal mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602806.png" /> into a ring under which all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602807.png" /> become invertible. However, there are explicit constructions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602808.png" />:
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1) as the set of fractions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l0602809.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028011.png" /> is a product of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028012.png" /> (two fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028014.png" /> are regarded as equivalent if and only if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028015.png" /> that is a product of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028016.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028017.png" />; fractions are added and multiplied by the usual rules);
+
{{TEX|auto}}
 +
{{TEX|done}}
  
2) as the quotient ring of the ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028019.png" />, with respect to the ideal generated by the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028021.png" />;
+
A transition from a commutative ring  $  A $
 +
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]])  $  A [ S  ^ {-} 1 ] $,  
 +
where  $  S $
 +
is a subset of  $  A $.  
 +
The ring  $  A [ S  ^ {-} 1 ] $
 +
can be defined as the solution of the problem of a universal mapping from  $  A $
 +
into a ring under which all elements of  $  S $
 +
become invertible. However, there are explicit constructions for  $  A [ S  ^ {-} 1 ] $:
  
3) as the [[Inductive limit|inductive limit]] of an inductive system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028022.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028024.png" /> runs through a naturally-ordered free commutative monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028025.png" />. All the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028026.png" /> are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028027.png" />, and the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028028.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028029.png" /> coincide with multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028030.png" />.
+
1) as the set of fractions of the form  $  a / s $,  
 +
where $  a \in A $
 +
and  $  s $
 +
is a product of elements of  $  S $(
 +
two fractions  $  a / s $
 +
and  $  a  ^  \prime  / s  ^  \prime  $
 +
are regarded as equivalent if and only if there is an  $  s  ^ {\prime\prime} $
 +
that is a product of elements of  $  S $
 +
and is such that  $  s  ^ {\prime\prime} ( s a ^  \prime  - s  ^  \prime  a ) = 0 $;
 +
fractions are added and multiplied by the usual rules);
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028031.png" /> is canonically mapped into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028032.png" /> and converts the latter into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028033.png" />-algebra. This mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028034.png" /> is injective if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028035.png" /> does not contain any divisor of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028036.png" />. On the other hand, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028037.png" /> contains a nilpotent element, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028038.png" />.
+
2) as the quotient ring of the ring of polynomials  $  A [ X _ {s} ] $,
 +
$  s \in S $,
 +
with respect to the ideal generated by the polynomials  $  s X _ {s} - 1 $,  
 +
$  s \in S $;
  
Without loss of generality the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028039.png" /> can be assumed to be closed with respect to products (such a set is known as multiplicative, or as a multiplicative system). In this case the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028040.png" /> is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028041.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028042.png" />. The most important examples of multiplicative systems are the following:
+
3) as the [[Inductive limit|inductive limit]] of an inductive system of  $  A $-
 +
modules  $  ( A _ {i} , \phi _ {ij} ) $,  
 +
where  $  i $
 +
runs through a naturally-ordered free commutative monoid  $  N  ^ {(} S) $.  
 +
All the  $  A _ {i} $
 +
are isomorphic to  $  A $,
 +
and the homomorphisms  $  \phi _ {ij} :  A _ {i} \rightarrow A _ {j} $
 +
with  $  j = i + n _ {1} s _ {1} + \dots + n _ {k} s _ {k} $
 +
coincide with multiplication by $  s _ {1} ^ {n _ {1} } {} \dots s _ {k} ^ {n _ {k} } \in A $.
  
a) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028043.png" /> of all powers of an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028044.png" />;
+
The ring  $  A $
 +
is canonically mapped into  $  A [ S  ^ {-} 1 ] $
 +
and converts the latter into an  $  A $-
 +
algebra. This mapping  $  A \rightarrow A [ S  ^ {-} 1 ] $
 +
is injective if and only if  $  S $
 +
does not contain any divisor of zero in  $  A $.  
 +
On the other hand, if  $  S $
 +
contains a nilpotent element, then  $  A [ S  ^ {-} 1 ] = 0 $.
  
b) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028045.png" />, that is, the complement of a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028046.png" />. The corresponding ring of fractions is local and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028047.png" />;
+
Without loss of generality the set $  S $
 +
can be assumed to be closed with respect to products (such a set is known as multiplicative, or as a multiplicative system). In this case the ring $  A [ S  ^ {-} 1 ] $
 +
is also denoted by $  S  ^ {-} 1 A $
 +
or  $  A _ {S} $.
 +
The most important examples of multiplicative systems are the following:
  
c) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028048.png" /> of all non-divisors of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028049.png" />.
+
a) the set $  \{ s  ^ {n} \} $
 +
of all powers of an element of  $  A $;
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028050.png" /> is called the complete ring of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028052.png" /> is integral, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028053.png" /> is a field of fractions.
+
b) the set  $  A \setminus  \mathfrak P $,
 +
that is, the complement of a prime ideal  $  \mathfrak P $.
 +
The corresponding ring of fractions is local and is denoted by  $  A _ {\mathfrak P }  $;
  
The operation of localization carries over with no difficulty to arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028054.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028055.png" /> if one sets
+
c) the set  $  R $
 +
of all non-divisors of zero in  $  A $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028056.png" /></td> </tr></table>
+
The ring  $  R  ^ {-} 1 A $
 +
is called the complete ring of fractions of  $  A $.
 +
If  $  A $
 +
is integral, then  $  R  ^ {-} 1 A = A _ {(} 0) $
 +
is a field of fractions.
  
The transition from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028057.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028058.png" /> is an exact functor. In other words, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028059.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028060.png" /> is flat. Localization commutes with direct sums and inductive limits.
+
The operation of localization carries over with no difficulty to arbitrary  $  A $-
 +
modules  $  M $
 +
if one sets
  
From the geometrical point of view localization means transition to an open subset. More precisely, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028061.png" /> the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028062.png" /> is canonically identified with the open (in the [[Zariski topology|Zariski topology]]) subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028063.png" /> consisting of the prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028064.png" /> not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028065.png" />. Moreover, this operation makes it possible to associate with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028066.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028067.png" /> a quasi-coherent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028068.png" /> on the affine scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028069.png" /> for which
+
$$
 +
M [ S  ^ {-} 1 ] = M \otimes _ {A} A [ S  ^ {-} 1 ] .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028070.png" /></td> </tr></table>
+
The transition from  $  M $
 +
to  $  M [ S  ^ {-} 1 ] $
 +
is an exact functor. In other words, the  $  A $-
 +
module  $  A [ S  ^ {-} 1 ] $
 +
is flat. Localization commutes with direct sums and inductive limits.
  
Localization can be regarded as an operation that makes it possible to invert morphisms of multiplication by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028071.png" /> in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060280/l06028072.png" />-modules. In this approach the operation of localization admits a wide generalization to arbitrary categories (see [[Localization in categories|Localization in categories]]).
+
From the geometrical point of view localization means transition to an open subset. More precisely, for  $  s \in A $
 +
the spectrum  $  \mathop{\rm Spec}  A [ s  ^ {-} 1 ] $
 +
is canonically identified with the open (in the [[Zariski topology|Zariski topology]]) subset  $  D ( s) \subset  \mathop{\rm Spec}  A $
 +
consisting of the prime ideals  $  \mathfrak P $
 +
not containing  $  s $.
 +
Moreover, this operation makes it possible to associate with each  $  A $-
 +
module  $  M $
 +
a quasi-coherent sheaf  $  \widetilde{M}  $
 +
on the affine scheme  $  \mathop{\rm Spec}  A $
 +
for which
 +
 
 +
$$
 +
\Gamma ( D ( s) , \widetilde{M}  )  =  M [ S  ^ {-} 1 ] .
 +
$$
 +
 
 +
Localization can be regarded as an operation that makes it possible to invert morphisms of multiplication by an $  s \in S $
 +
in the category of $  A $-
 +
modules. In this approach the operation of localization admits a wide generalization to arbitrary categories (see [[Localization in categories|Localization in categories]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>

Revision as of 22:17, 5 June 2020


A transition from a commutative ring $ A $ to the ring of fractions (cf. Fractions, ring of) $ A [ S ^ {-} 1 ] $, where $ S $ is a subset of $ A $. The ring $ A [ S ^ {-} 1 ] $ can be defined as the solution of the problem of a universal mapping from $ A $ into a ring under which all elements of $ S $ become invertible. However, there are explicit constructions for $ A [ S ^ {-} 1 ] $:

1) as the set of fractions of the form $ a / s $, where $ a \in A $ and $ s $ is a product of elements of $ S $( two fractions $ a / s $ and $ a ^ \prime / s ^ \prime $ are regarded as equivalent if and only if there is an $ s ^ {\prime\prime} $ that is a product of elements of $ S $ and is such that $ s ^ {\prime\prime} ( s a ^ \prime - s ^ \prime a ) = 0 $; fractions are added and multiplied by the usual rules);

2) as the quotient ring of the ring of polynomials $ A [ X _ {s} ] $, $ s \in S $, with respect to the ideal generated by the polynomials $ s X _ {s} - 1 $, $ s \in S $;

3) as the inductive limit of an inductive system of $ A $- modules $ ( A _ {i} , \phi _ {ij} ) $, where $ i $ runs through a naturally-ordered free commutative monoid $ N ^ {(} S) $. All the $ A _ {i} $ are isomorphic to $ A $, and the homomorphisms $ \phi _ {ij} : A _ {i} \rightarrow A _ {j} $ with $ j = i + n _ {1} s _ {1} + \dots + n _ {k} s _ {k} $ coincide with multiplication by $ s _ {1} ^ {n _ {1} } {} \dots s _ {k} ^ {n _ {k} } \in A $.

The ring $ A $ is canonically mapped into $ A [ S ^ {-} 1 ] $ and converts the latter into an $ A $- algebra. This mapping $ A \rightarrow A [ S ^ {-} 1 ] $ is injective if and only if $ S $ does not contain any divisor of zero in $ A $. On the other hand, if $ S $ contains a nilpotent element, then $ A [ S ^ {-} 1 ] = 0 $.

Without loss of generality the set $ S $ can be assumed to be closed with respect to products (such a set is known as multiplicative, or as a multiplicative system). In this case the ring $ A [ S ^ {-} 1 ] $ is also denoted by $ S ^ {-} 1 A $ or $ A _ {S} $. The most important examples of multiplicative systems are the following:

a) the set $ \{ s ^ {n} \} $ of all powers of an element of $ A $;

b) the set $ A \setminus \mathfrak P $, that is, the complement of a prime ideal $ \mathfrak P $. The corresponding ring of fractions is local and is denoted by $ A _ {\mathfrak P } $;

c) the set $ R $ of all non-divisors of zero in $ A $.

The ring $ R ^ {-} 1 A $ is called the complete ring of fractions of $ A $. If $ A $ is integral, then $ R ^ {-} 1 A = A _ {(} 0) $ is a field of fractions.

The operation of localization carries over with no difficulty to arbitrary $ A $- modules $ M $ if one sets

$$ M [ S ^ {-} 1 ] = M \otimes _ {A} A [ S ^ {-} 1 ] . $$

The transition from $ M $ to $ M [ S ^ {-} 1 ] $ is an exact functor. In other words, the $ A $- module $ A [ S ^ {-} 1 ] $ is flat. Localization commutes with direct sums and inductive limits.

From the geometrical point of view localization means transition to an open subset. More precisely, for $ s \in A $ the spectrum $ \mathop{\rm Spec} A [ s ^ {-} 1 ] $ is canonically identified with the open (in the Zariski topology) subset $ D ( s) \subset \mathop{\rm Spec} A $ consisting of the prime ideals $ \mathfrak P $ not containing $ s $. Moreover, this operation makes it possible to associate with each $ A $- module $ M $ a quasi-coherent sheaf $ \widetilde{M} $ on the affine scheme $ \mathop{\rm Spec} A $ for which

$$ \Gamma ( D ( s) , \widetilde{M} ) = M [ S ^ {-} 1 ] . $$

Localization can be regarded as an operation that makes it possible to invert morphisms of multiplication by an $ s \in S $ in the category of $ A $- modules. In this approach the operation of localization admits a wide generalization to arbitrary categories (see Localization in categories).

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
How to Cite This Entry:
Localization in a commutative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Localization_in_a_commutative_algebra&oldid=47687
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article