Glossary

5.2 Localization at prime ideals and local rings

2 min readโ€ขjuly 25, 2024

Localization at prime ideals is a powerful tool in commutative algebra. It allows us to zoom in on specific parts of a ring, making complex problems more manageable by focusing on local properties.

This technique creates a new ring where elements outside the prime ideal become units. It establishes a useful correspondence between prime ideals in the original ring and its localization, simplifying many ring-theoretic problems.

Localization at Prime Ideals

Localization of rings at prime ideals

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  • Localization of ring R at prime ideal p denoted Rp=Sโˆ’1RR_p = S^{-1}R, S comprises elements not in p
  • Elements in RpR_p expressed as fractions a/sa/s (a in R, s in S) representing equivalence classes
  • Localization preserves commutativity and unity of original ring
  • Fractions a/s=b/ta/s = b/t equivalent when (atโˆ’bs)u=0(at - bs)u = 0 for some u in S
  • Natural homomorphism ฯ•:Rโ†’Rp\phi: R \to R_p maps r to r/1
  • Kernel of ฯ•\phi contains elements r where rs = 0 for some s in S

Local rings and prime ideal localization

  • Local rings characterized by single (fields, formal power series rings)
  • Non-units in form
  • Sum of non-units remains non-unit in local rings
  • Localization RpR_p forms local ring with maximal ideal pRp={a/s:aโˆˆp,sโˆˆS}pR_p = \{a/s : a \in p, s \in S\}

Localization as local ring proof

  1. Establish pRppR_p as ideal in RpR_p
  2. Demonstrate maximality of pRppR_p in RpR_p
  3. Prove uniqueness of pRppR_p as maximal ideal
  • Elements outside pRppR_p become units in RpR_p
  • All proper ideals of RpR_p contained within pRppR_p

Prime ideals in rings vs localizations

  • Bijective correspondence between prime ideals of RpR_p and prime ideals of R contained in p
  • Order-preserving relationship: Q1โŠ‚Q2Q_1 \subset Q_2 in RpR_p if and only if q1โŠ‚q2q_1 \subset q_2 in R
  • Prime ideals strictly inside p remain prime in RpR_p
  • Prime ideals outside p become unit ideal in RpR_p
  • Localization technique simplifies ring-theoretic problems by focusing on local properties

Key Terms to Review (16)

Affine Schemes: Affine schemes are the basic building blocks of algebraic geometry, defined as the spectrum of a commutative ring, denoted as Spec(A), where A is a ring. They capture the geometric properties of algebraic varieties and allow for a robust connection between algebra and geometry. An affine scheme represents the set of prime ideals of the ring, and its structure sheaf encodes the algebraic functions that can be defined on it.
Completion of a local ring: The completion of a local ring is a construction that allows us to form a new ring that captures the 'limit' behavior of sequences of elements in the local ring, particularly with respect to its maximal ideal. This process is essential for understanding properties of local rings, especially when studying their behavior under various topological perspectives and analyzing their structure through formal power series.
Flatness: Flatness is a property of a module over a ring, indicating that the module behaves like a 'flat' version of free modules in the sense that the functor of tensoring with it preserves exact sequences. This property is crucial when dealing with localization and tensor products, as flat modules allow for the smooth transition between various algebraic structures without introducing new relations or losing information.
Krull dimension: Krull dimension is a fundamental concept in commutative algebra that measures the 'size' of a ring by considering the maximum length of chains of prime ideals. This dimension helps to understand the structure of rings and their prime ideals, which connects to various important properties and theorems in algebraic geometry and ring theory.
Local coordinate ring: A local coordinate ring is a specific type of ring associated with a point on an algebraic variety, capturing the behavior of the variety near that point. This ring allows us to study local properties of the variety by focusing on functions that can be defined around that point, effectively providing a localized perspective on its geometric structure. The local coordinate ring plays a crucial role in connecting algebraic and geometric concepts, particularly in the context of prime ideals and local rings.
Local homomorphism: A local homomorphism is a type of ring homomorphism between two local rings where the image of the maximal ideal of the source ring is contained within the maximal ideal of the target ring. This concept is crucial in understanding how properties of local rings behave under homomorphisms and emphasizes how localization can preserve certain algebraic structures.
Local ring: A local ring is a ring that has a unique maximal ideal, which allows us to focus on the behavior of functions and elements near a certain point. This concept is crucial for understanding properties like prime ideals and localization, as it helps isolate the study of algebraic structures around specific elements, making it easier to analyze their local behavior and properties.
Localization at a prime ideal: Localization at a prime ideal is a process in commutative algebra that focuses on a particular prime ideal in a ring, allowing mathematicians to create a new ring where the elements of that prime ideal become 'inverted' or non-zero. This process helps analyze the properties of the original ring in a more manageable way, especially near the prime ideal, leading to concepts like local rings and providing insights into the structure of algebraic varieties. The resulting local ring retains many useful properties and serves as a crucial tool for various applications in algebraic geometry and number theory.
Maximal ideal: A maximal ideal is an ideal in a ring that is proper (not equal to the entire ring) and has the property that there are no other ideals containing it except for itself and the entire ring. These ideals play a crucial role in understanding the structure of rings, especially in relation to fields and quotient rings.
Nakayama's Lemma: Nakayama's Lemma is a fundamental result in commutative algebra that deals with the relationships between ideals and modules over local rings. It essentially states that if you have a finitely generated module over a local ring, then if the module is annihilated by a certain ideal, it can be shown that the module must be zero. This lemma has deep implications in understanding the structure of local rings and helps simplify many problems in algebraic geometry and algebraic number theory.
Prime Spectrum: The prime spectrum of a ring is the set of all prime ideals in that ring, often denoted as \(\text{Spec}(R)\). This concept is crucial in understanding the structure of rings, as it reveals how prime ideals relate to various properties of the ring, such as its decompositions and factorization. The prime spectrum provides insight into localization processes and the relationships between Artinian and Noetherian rings, highlighting their ideal structures and behaviors.
Ring of Formal Power Series: The ring of formal power series is a set of sequences of coefficients indexed by non-negative integers, equipped with operations of addition and multiplication defined in a manner similar to polynomials. This ring allows for manipulation of infinite series as if they were polynomials, providing a powerful tool in various areas of mathematics, including localization at prime ideals and local rings. The elements of this ring can be thought of as formal sums that can converge in certain contexts, making them useful for studying local properties of algebraic structures.
Total ring of fractions: The total ring of fractions is a construction that generalizes the idea of localization by allowing for inverting all non-zero elements of a given ring. This approach is especially useful when dealing with rings that are not necessarily integral domains, enabling the study of their properties in a broader context. In the total ring of fractions, elements are expressed as equivalence classes of fractions, where the denominators are non-zero elements, facilitating a comprehensive analysis of the ring's structure and its relationships with local rings.
Unique Maximal Ideal: A unique maximal ideal is an ideal in a ring that is maximal among all proper ideals, meaning there are no larger ideals contained within it, and it is the only one of its kind in that context. This concept is closely tied to local rings, where the existence of a unique maximal ideal allows for localization, which in turn creates a setting where algebraic properties can be examined more easily. The significance of having a unique maximal ideal plays a crucial role in understanding the structure of local rings and their applications in algebraic geometry and commutative algebra.
Zariski topology: Zariski topology is a mathematical structure that defines a topology on the spectrum of a commutative ring, particularly focusing on prime ideals. It allows us to associate algebraic sets with geometric concepts by treating the prime ideals as points in a space and the vanishing sets of polynomials as closed sets. This topology provides a way to study algebraic varieties through their coordinate rings and connects algebraic geometry with commutative algebra.
Zariski's Lemma: Zariski's Lemma is a fundamental result in commutative algebra that provides a connection between local properties of rings and their prime ideals. It essentially states that for any prime ideal in a Noetherian ring, the localization at that prime ideal behaves well, particularly in terms of finitely generated modules. This lemma is crucial for understanding local rings, which allow us to study algebraic varieties and their points in a localized context.
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