Localization in a commutative algebra

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)