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Difference between revisions of "Localization in a commutative algebra"

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A transition from a commutative ring  $  A $
 
A transition from a commutative ring  $  A $
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]])  $  A [ S  ^ {-} 1 ] $,  
+
to the ring of fractions (cf. [[Fractions, ring of|Fractions, ring of]])  $  A [ S  ^ {-1} ] $,  
 
where  $  S $
 
where  $  S $
 
is a subset of  $  A $.  
 
is a subset of  $  A $.  
The ring  $  A [ S  ^ {-} 1 ] $
+
The ring  $  A [ S  ^ {-1} ] $
 
can be defined as the solution of the problem of a universal mapping from  $  A $
 
can be defined as the solution of the problem of a universal mapping from  $  A $
 
into a ring under which all elements of  $  S $
 
into a ring under which all elements of  $  S $
become invertible. However, there are explicit constructions for  $  A [ S  ^ {-} 1 ] $:
+
become invertible. However, there are explicit constructions for  $  A [ S  ^ {-1} ] $:
  
 
1) as the set of fractions of the form  $  a / s $,  
 
1) as the set of fractions of the form  $  a / s $,  
Line 39: Line 39:
 
modules  $  ( A _ {i} , \phi _ {ij} ) $,  
 
modules  $  ( A _ {i} , \phi _ {ij} ) $,  
 
where  $  i $
 
where  $  i $
runs through a naturally-ordered free commutative monoid  $  N  ^ {(} S) $.  
+
runs through a naturally-ordered free commutative monoid  $  N  ^ {(S)} $.  
 
All the  $  A _ {i} $
 
All the  $  A _ {i} $
 
are isomorphic to  $  A $,  
 
are isomorphic to  $  A $,  
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The ring  $  A $
 
The ring  $  A $
is canonically mapped into  $  A [ S  ^ {-} 1 ] $
+
is canonically mapped into  $  A [ S  ^ {-1} ] $
and converts the latter into an  $  A $-
+
and converts the latter into an  $  A $-algebra. This mapping  $  A \rightarrow A [ S  ^ {-1} ] $
algebra. This mapping  $  A \rightarrow A [ S  ^ {-} 1 ] $
 
 
is injective if and only if  $  S $
 
is injective if and only if  $  S $
 
does not contain any divisor of zero in  $  A $.  
 
does not contain any divisor of zero in  $  A $.  
 
On the other hand, if  $  S $
 
On the other hand, if  $  S $
contains a nilpotent element, then  $  A [ S  ^ {-} 1 ] = 0 $.
+
contains a nilpotent element, then  $  A [ S  ^ {-1} ] = 0 $.
  
 
Without loss of generality the set  $  S $
 
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 ] $
+
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 $
+
is also denoted by  $  S  ^ {-1} A $
 
or  $  A _ {S} $.  
 
or  $  A _ {S} $.  
 
The most important examples of multiplicative systems are the following:
 
The most important examples of multiplicative systems are the following:
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of all non-divisors of zero in  $  A $.
 
of all non-divisors of zero in  $  A $.
  
The ring  $  R  ^ {-} 1 A $
+
The ring  $  R  ^ {-1} A $
 
is called the complete ring of fractions of  $  A $.  
 
is called the complete ring of fractions of  $  A $.  
 
If  $  A $
 
If  $  A $
is integral, then  $  R  ^ {-} 1 A = A _ {(} 0) $
+
is integral, then  $  R  ^ {-1} A = A _ {(0)} $
 
is a field of fractions.
 
is a field of fractions.
  
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$$  
 
$$  
M [ S  ^ {-} 1 ]  =  M \otimes _ {A} A [ S  ^ {-} 1 ] .
+
M [ S  ^ {-1} ]  =  M \otimes _ {A} A [ S  ^ {-1} ] .
 
$$
 
$$
  
 
The transition from  $  M $
 
The transition from  $  M $
to  $  M [ S  ^ {-} 1 ] $
+
to  $  M [ S  ^ {-1} ] $
 
is an exact functor. In other words, the  $  A $-
 
is an exact functor. In other words, the  $  A $-
module  $  A [ S  ^ {-} 1 ] $
+
module  $  A [ S  ^ {-1} ] $
 
is flat. Localization commutes with direct sums and inductive limits.
 
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 $
 
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 ] $
+
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 $
 
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 $
 
consisting of the prime ideals  $  \mathfrak P $
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$$  
 
$$  
\Gamma ( D ( s) , \widetilde{M}  )  =  M [ S  ^ {-} 1 ] .
+
\Gamma ( D ( s) , \widetilde{M}  )  =  M [ S  ^ {-1} ] .
 
$$
 
$$
  

Latest revision as of 21:35, 4 January 2021


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