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Localization


An operation on rings and modules. Given a commutative unit ring R, and a subset S of R, closed under multiplication, such that 1 in S, and 0 not in S, the localization of R at S is the ring

 R_S={a/s|a in R,s in S},
(1)

where the addition and the multiplication of the formal fractions a/s are defined according to the natural rules,

 a/s+b/t=(at+bs)/(st),
(2)

and

 a/s·b/t=(ab)/(st).
(3)

The ring R is a subring of R_S via the identification a=a/1.

For an R-module M, the localization of M at S is defined as the tensor product M tensor _RR_S, i.e., as the set of linear combinations of the elementary tensors

 m tensor 1/s(m in M,s in S),
(4)

which are also denoted m/s for short.

The properties required for the subset S are fulfilled by

1. The set of non zero-divisors of R; in this case R_S is the ring of fractions of R.

2. The complement R\P of any prime ideal P of R: in this case the clumsy notation R_(R\P) is replaced by R_P. This ring is called the localization of R at P, and it is a local ring, with maximal ideal PR_P.

The name given to this operation derives from the geometric meaning it takes when applied to the rings associated with algebraic varieties.

The union V of the coordinate axes of the real Cartesian plane is an algebraic variety given by the equation

 XY=0
(5)

and is associated with the quotient ring R=R[X,Y]/<XY>. The localization of R at the maximal ideal generated by the residues x of X and y of Y, denoted R_(<x,y>), describes V at the origin; its algebraic properties provide clues to local geometric properties. For example, the localized ring is nonregular since its Krull dimension is 1, whereas two elements are needed to generate its maximal ideal <x,y>R_(<x,y>). This is the algebraic counterpart of the fact that the origin is a singular point (a knot) for V. For all other points P(alpha,0) of the X-axis, (alpha!=0), the localized ring R_(<x-alpha,y>) is regular of dimension 1, since x-alpha generates the whole maximal ideal <x-alpha,y>R_(<x-alpha,y>): being xy=0, one has that y=-(1/alpha)y(x-alpha). The same argument applies to the Y-axis. It follows that outside the origin variety V is regular in the geometric meaning of "smooth" or "nonsingular."


See also

Local Ring, Regular Local Ring, Singular Point

This entry contributed by Margherita Barile

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References

Bruns, W. and Herzog, J. Cohen-Macaulay Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.Matsumura, H. Commutative Ring Theory. Cambridge, England: Cambridge University Press, 1986.Nagata, M. Local Rings. Huntington, NY: Krieger, 1975.Samuel, P. and Zariski, O. Commutative Algebra II. Princeton, NJ: Van Nostrand, 1958.

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Localization

Cite this as:

Barile, Margherita. "Localization." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Localization.html

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