Localization is a powerful tool in commutative algebra that zooms in on specific parts of a ring. By focusing on a subset of elements, we can study local properties and simplify complex structures.
Local rings, with their unique , are key players in algebraic geometry. They help us understand the behavior of algebraic varieties near specific points, bridging the gap between algebra and geometry.
Localization of rings
Definition and notation
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The R at a multiplicative subset S, denoted S−1R, is the ring of fractions with denominators in S
Elements in S−1R are of the form r/s where r∈R and s∈S
The localization of a ring R at a prime ideal p, denoted Rp, is the localization of R at the multiplicative set R−p
Elements in Rp are of the form r/s where r∈R and s∈/p
The localization of a ring R at a maximal ideal m, denoted Rm, is called the at m
The localization of a ring R at the multiplicative set {1,f,f2,...} for some f∈R is denoted Rf
Construction and properties
The localization S−1R is constructed as the set of equivalence classes of pairs (r,s) with r∈R and s∈S, where (r1,s1)∼(r2,s2) if there exists t∈S such that t(s1r2−s2r1)=0
The equivalence class of (r,s) in S−1R is denoted by r/s
Addition and multiplication in S−1R are defined by (r1/s1)+(r2/s2)=(s2r1+s1r2)/(s1s2) and (r1/s1)(r2/s2)=(r1r2)/(s1s2)
The localization S−1R is a ring with identity element 1/1
The natural map ϕ:R→S−1R given by r↦r/1 is a
ϕ is injective if and only if S contains no zero divisors
If R is an and S=R−{0}, then S−1R is the of R (e.g., Q is the field of fractions of Z)
Properties of localization
Ideals and prime ideals
If I is an ideal of R, then S−1I={i/s:i∈I,s∈S} is an ideal of S−1R
The map I↦S−1I gives a bijection between the ideals of R that do not intersect S and the ideals of S−1R
If p is a prime ideal of R, then pRp is the unique maximal ideal of Rp
The map p↦pRp gives a bijection between the prime ideals of R that do not intersect S and the prime ideals of S−1R
Relationship between a ring and its localizations
For any multiplicative subset S of R, the ring R can be viewed as a subring of S−1R via the natural map ϕ:R→S−1R
The localization S−1R can be viewed as a "local version" of R where elements outside of S are inverted
This allows for the study of local properties of R (e.g., at a specific prime ideal)
Local rings and examples
Definition and properties
A local ring is a ring with a unique maximal ideal
The localization of a ring R at a prime ideal p, denoted Rp, is a local ring with maximal ideal pRp
In a local ring (R,m), every element not in m is a unit (invertible)
This is because m is the only maximal ideal, so any proper ideal is contained in m
Examples of local rings
The ring of germs of continuous functions at a point on a topological space is a local ring
The ring of convergent power series over a field is a local ring
e.g., R[[x]], the ring of formal power series with real coefficients
The ring of rational functions on an algebraic variety, localized at a point, is a local ring
e.g., k[x,y](x,y), the localization of the polynomial ring k[x,y] at the maximal ideal (x,y)
Examples of non-local rings
The ring of integers Z is not a local ring, as it has infinitely many maximal ideals (one for each prime number)
The ring of polynomials k[x] over a field k is not a local ring, as it has infinitely many maximal ideals (one for each irreducible polynomial)
However, localizing k[x] at a specific maximal ideal (e.g., (x−a) for some a∈k) yields a local ring
Ring vs localization relationship
Injective ring homomorphism
The natural map ϕ:R→S−1R is an injective ring homomorphism if and only if S contains no zero divisors
If S contains a zero divisor s, then ϕ(s)=s/1 is a zero divisor in S−1R, contradicting injectivity
Conversely, if S contains no zero divisors and ϕ(r)=0, then r/1=0/1, implying tr=0 for some t∈S, which forces r=0 since t is not a zero divisor
Correspondence between ideals
The map I↦S−1I gives a bijection between the ideals of R that do not intersect S and the ideals of S−1R
If I∩S=∅, then S−1I=S−1R, which corresponds to the improper ideal of S−1R
The map p↦pRp gives a bijection between the prime ideals of R that do not intersect S and the prime ideals of S−1R
This bijection preserves inclusions, i.e., if p⊆q, then pRp⊆qRq
Localization as a subring
For any multiplicative subset S of R, the ring R can be viewed as a subring of S−1R via the natural map ϕ:R→S−1R
This embedding allows for the transfer of properties from R to S−1R and vice versa
For example, if R is Noetherian, then so is S−1R; if S−1R is an integral domain, then so is R