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
Key Terms to Review (16)
Field of Fractions: A field of fractions is a construction that allows one to create a field from an integral domain by introducing formal ratios of its elements. This concept is essential in algebra as it enables the extension of the set of numbers we can work with, particularly when dealing with polynomial rings and their local properties. It connects directly to how we can analyze algebraic structures in both affine varieties and local rings, where we often need to consider quotients of elements to understand their behaviors better.
Going Up Theorem: The Going Up Theorem is a fundamental result in commutative algebra that describes the behavior of prime ideals under localization. Specifically, it states that if you have a ring and a prime ideal, the prime ideal remains prime after localizing at a multiplicative set of the ring. This theorem highlights the relationship between the structure of rings and their localizations, emphasizing how properties of ideals can change or remain intact through this process.
Integral Domain: An integral domain is a type of commutative ring with unity that has no zero divisors, meaning that if the product of two non-zero elements is zero, then at least one of the elements must be zero. This property ensures that the cancellation law holds, making it a vital structure in algebraic settings. Integral domains provide a framework for defining concepts like primes and irreducible elements, which are crucial when studying localization and local rings.
Isomorphism: An isomorphism is a mathematical concept that establishes a one-to-one correspondence between two structures, demonstrating that they are fundamentally the same in terms of their properties and operations. In algebraic contexts, isomorphisms reveal deep connections between different algebraic objects, allowing us to treat them as interchangeable in certain aspects. This concept plays a vital role in understanding both local rings through localization and the structure of elliptic curves over finite fields.
Local Ring: A local ring is a commutative ring with a unique maximal ideal, making it an essential structure in algebraic geometry and commutative algebra. This property allows us to focus on the behavior of functions or elements near a specific point or prime ideal, which is crucial for studying local properties of varieties and schemes. The local ring captures the essence of the geometry at that point, facilitating the analysis of singularities, dimensions, and other local features.
Localization at a prime ideal: Localization at a prime ideal is a process in commutative algebra where one focuses on a specific prime ideal in a ring, allowing one to study the behavior of elements and properties of the ring in the vicinity of that ideal. This process transforms the ring into a local ring, where certain elements are inverted, enabling a more detailed analysis of properties like spectra and morphisms. This localized perspective helps in understanding how algebraic structures behave locally around points defined by prime ideals.
Localization of a Ring: Localization of a ring is a process in algebra that allows us to create a new ring from an existing one by inverting certain elements. This technique helps focus on properties of the ring that are important in a specific context, particularly useful for studying local properties of schemes. By localizing, we can simplify problems and analyze behavior around particular points or subsets, making it an essential tool in algebraic geometry.
Localizing an algebraic variety: Localizing an algebraic variety involves examining the variety in a smaller, more focused context by restricting attention to a particular point or open set. This process allows for the study of local properties, enabling a deeper understanding of the structure and behavior of the variety around that specific location, often leading to insights about global features. In algebraic geometry, localization is closely linked to local rings and provides a framework for analyzing singularities and other important characteristics of varieties.
Lying Over Theorem: The Lying Over Theorem states that if a ring homomorphism is surjective, then any prime ideal of the target ring lying over a prime ideal of the source ring has a corresponding prime ideal that lies beneath it in the source ring. This theorem is crucial for understanding how the properties of rings are preserved under localization and plays a significant role in the study of local rings.
Maximal ideal: A maximal ideal is a proper ideal of a ring that is not contained in any other proper ideal of that ring. This means that if you have an ideal that is maximal, any ideal that contains it is either the whole ring or the maximal ideal itself. Maximal ideals are crucial for understanding the structure of rings and play an important role in both localization processes and in the study of modules, where they help define points in algebraic geometry through their correspondence with geometric points in varieties.
Power Series Ring: A power series ring is a type of mathematical structure formed by formal expressions of the form $$ ext{a}_0 + ext{a}_1 x + ext{a}_2 x^2 + ext{a}_3 x^3 + ...$$ where $$ ext{a}_i$$ are coefficients from a ring and $$x$$ is an indeterminate. This structure allows for the manipulation and study of series that can converge under certain conditions, making it a powerful tool in algebraic geometry and local ring theory.
Property of being a local ring: The property of being a local ring refers to the characteristic of a ring that has a unique maximal ideal. This unique maximal ideal is essential as it allows us to focus on the behavior of functions and algebraic structures in a 'local' context. Local rings are particularly significant in algebraic geometry, where they help analyze the properties of varieties at specific points, making them invaluable for understanding singularities and local behaviors of geometric objects.
Ring Homomorphism: A ring homomorphism is a function between two rings that preserves the ring operations, namely addition and multiplication. This means that if you have two rings, R and S, a ring homomorphism `f: R → S` satisfies `f(a + b) = f(a) + f(b)` and `f(ab) = f(a)f(b)` for all elements a, b in R. This concept connects to important structures in algebra, particularly when discussing ideals, modules, and the properties of different types of rings.
Spectrum of a ring: The spectrum of a ring, denoted as Spec(R), is the set of all prime ideals of a commutative ring R, along with a Zariski topology that makes it a topological space. This concept connects algebra and geometry, allowing us to study algebraic varieties through their coordinate rings. By exploring the prime ideals, we gain insights into the structure of the ring and can understand the relationships between geometric objects and their algebraic counterparts.
Uniqueness of Maximal Ideal: The uniqueness of maximal ideal refers to the property of a local ring where its maximal ideal is the only maximal ideal present, making it unique. This concept is significant because it highlights how local rings focus on the behavior of functions around a single point, effectively capturing the essence of algebraic structures in that neighborhood. When dealing with localization, understanding that a local ring can have just one maximal ideal simplifies many aspects of algebraic geometry and algebraic structures.
Zariski topology: Zariski topology is a mathematical structure that defines a topology on algebraic varieties by considering the closed sets to be defined by polynomial equations. This topology is particularly useful in algebraic geometry as it allows for the study of geometric properties of solutions to polynomial equations. The closed sets correspond to the zero sets of collections of polynomials, leading to significant connections with affine and projective schemes, localization, Noetherian rings, and the theory of affine varieties.