1.5 Dedekind domains

Dedekind domains are key players in arithmetic geometry, bridging abstract algebra and number theory. They provide a framework for studying algebraic number fields, offering insights into algebraic integers and ideal factorization.

These domains generalize unique factorization from elements to ideals, a crucial concept in understanding algebraic structures. By exploring Dedekind domains, we gain powerful tools for analyzing number fields and their arithmetic properties.

Definition of Dedekind domains

• Dedekind domains form a crucial foundation in arithmetic geometry, bridging abstract algebra and number theory
• These domains provide a framework for studying algebraic number fields and their properties
• Understanding Dedekind domains enables deeper insights into the structure of algebraic integers and ideal factorization

Properties of Dedekind domains

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• Noetherian rings characterized by every non-zero proper ideal factoring uniquely into prime ideals
• Integrally closed domains ensuring algebraic completeness within their field of fractions
• Dimension one rings with all non-zero prime ideals maximal
• Possess unique factorization of ideals, generalizing unique factorization in principal ideal domains
• Every non-zero fractional ideal invertible, leading to a rich theory of ideal arithmetic

Examples of Dedekind domains

• Rings of integers in algebraic number fields ($\mathbb{Z}[\sqrt{-5}]$)
• Polynomial rings over fields in one variable ($k[x]$ where $k$ is a field)
• Coordinate rings of non-singular algebraic curves over algebraically closed fields
• Discrete valuation rings with unique maximal ideal ($\mathbb{Z}_{(p)}$, the localization of $\mathbb{Z}$ at a prime $p$)

Non-examples of Dedekind domains

• Polynomial rings in two or more variables over fields ($k[x,y]$)
• Rings with zero divisors ($\mathbb{Z}/6\mathbb{Z}$)
• Non-integrally closed domains ($\mathbb{Z}[\sqrt{-3}]$)
• Rings of continuous functions on connected spaces
• Fields, as they lack proper non-zero ideals

Ideal theory in Dedekind domains

• Ideal theory in Dedekind domains extends the concept of factorization from elements to ideals
• This generalization allows for a deeper understanding of algebraic structures in arithmetic geometry
• Studying ideals in Dedekind domains provides insights into the arithmetic of algebraic number fields

Prime ideals

• Non-zero prime ideals in Dedekind domains always maximal
• Generate prime ideal spectrum of height one
• Correspond to prime divisors in algebraic number theory
• Play a crucial role in the unique factorization of ideals
• Can be used to define valuations on the fraction field

Maximal ideals

• Coincide with non-zero prime ideals in Dedekind domains
• Generate residue fields which are finite extensions of the base field
• Correspond to closed points in the associated affine scheme
• Used to define local rings and study local properties
• Determine the Jacobson radical of the domain

Principal ideals

• Generated by a single element of the domain
• May not always exist for every ideal in a Dedekind domain
• Measure how far a Dedekind domain deviates from being a PID
• Related to the ideal class group of the domain
• Can be used to study divisibility properties in the domain

Fractional ideals

• Submodules of the fraction field containing a non-zero element of the domain
• Form a group under multiplication, with integer ideals as a submonoid
• Invertible, allowing for a rich theory of ideal arithmetic
• Used to define the ideal class group of a Dedekind domain
• Provide a geometric interpretation of divisors on algebraic curves

Factorization in Dedekind domains

• Factorization in Dedekind domains generalizes unique factorization from elements to ideals
• This concept plays a central role in arithmetic geometry, particularly in studying algebraic number fields
• Understanding factorization in Dedekind domains provides insights into the arithmetic of algebraic integers

Unique factorization of ideals

• Every non-zero ideal factors uniquely as a product of prime ideals
• Generalizes unique factorization of elements in UFDs to ideals
• Allows for a systematic study of divisibility in Dedekind domains
• Provides a powerful tool for analyzing the structure of ideals
• Leads to important applications in algebraic number theory (class number, ideal class group)

Prime factorization theorem

• States that every non-zero ideal $I$ can be written uniquely as $I = \mathfrak{p}_1^{e_1} \cdots \mathfrak{p}_n^{e_n}$
• $\mathfrak{p}_i$ represent distinct prime ideals and $e_i$ positive integers
• Exponents $e_i$ called the valuations of $I$ at $\mathfrak{p}_i$
• Analogous to prime factorization of integers, but for ideals
• Fundamental for studying ramification in algebraic number fields

Factorization vs principal ideals

• Not all ideals in a Dedekind domain principal, unlike in PIDs
• Measure of non-principal ideals given by the ideal class group
• Principal ideals correspond to divisors of elements in the domain
• Non-principal ideals arise from more complex algebraic structures
• Study of this relationship leads to important invariants in algebraic number theory (class number)

Dedekind domains vs UFDs

• Comparison between Dedekind domains and Unique Factorization Domains (UFDs) crucial in arithmetic geometry
• Understanding their relationship provides insights into the structure of various algebraic objects
• This comparison highlights the generalization of factorization from elements to ideals

Similarities and differences

• Both Dedekind domains and UFDs have unique factorization, but in different contexts
• UFDs have unique factorization of elements, Dedekind domains of ideals
• Dedekind domains integrally closed, UFDs not necessarily
• All UFDs Dedekind domains in dimension one, but not conversely
• Dedekind domains may have non-principal ideals, UFDs only principal ideals

Relationship to PIDs

• Principal Ideal Domains (PIDs) form the intersection of Dedekind domains and UFDs
• Every PID a Dedekind domain, but not every Dedekind domain a PID
• PIDs have unique factorization of both elements and ideals
• Dedekind domains generalize PIDs by allowing non-principal ideals
• Study of when a Dedekind domain a PID leads to class field theory

Dedekind domains in number theory

• Dedekind domains play a fundamental role in algebraic number theory within arithmetic geometry
• They provide a framework for studying the arithmetic of algebraic number fields
• Understanding Dedekind domains in this context leads to deep results in class field theory and beyond

Rings of integers

• Integral closures of $\mathbb{Z}$ in finite extensions of $\mathbb{Q}$
• Always Dedekind domains, generalizing properties of $\mathbb{Z}$
• Provide a natural setting for studying algebraic integers
• Allow for the development of a rich theory of ideal factorization
• Key to understanding ramification and splitting of primes in number fields

Ideal class group

• Measures how far a Dedekind domain from being a PID
• Defined as the quotient of fractional ideals by principal fractional ideals
• Finite for rings of integers in number fields (class number)
• Trivial if and only if the Dedekind domain a PID
• Crucial in class field theory and the study of quadratic forms

Applications to algebraic number fields

• Dedekind domains used to study factorization of primes in field extensions
• Provide a framework for understanding ramification and splitting of primes
• Allow for the development of local-global principles in number theory
• Used in the study of Diophantine equations and arithmetic of elliptic curves
• Fundamental in the formulation and proof of the Hilbert class field theorem

Localization of Dedekind domains

• Localization of Dedekind domains a powerful technique in arithmetic geometry
• Allows for the study of local properties of algebraic varieties and number fields
• Provides a bridge between global and local aspects of arithmetic structures

Properties of localizations

• Localizations of Dedekind domains at prime ideals yield discrete valuation rings
• Preserve many properties of Dedekind domains (noetherian, integrally closed)
• Allow for the study of local behavior of ideals and elements
• Provide a tool for gluing local information to obtain global results
• Used in the construction of adeles and ideles in global class field theory

Discrete valuation rings

• Special case of Dedekind domains with unique non-zero prime ideal
• Characterized by having a uniformizing element generating the maximal ideal
• Provide a local model for studying singularities of algebraic curves
• Used in the theory of places of function fields
• Play a crucial role in ramification theory and local class field theory

Modules over Dedekind domains

• Study of modules over Dedekind domains central to understanding algebraic structures in arithmetic geometry
• Provides insights into the arithmetic of number fields and function fields
• Generalizes the theory of finitely generated abelian groups

Structure theorem

• Finitely generated modules over Dedekind domains decompose into free and torsion parts
• Torsion part further decomposes as direct sum of cyclic modules
• Generalizes the fundamental theorem of finitely generated abelian groups
• Allows for a classification of finitely generated modules up to isomorphism
• Crucial in the study of Galois modules and arithmetic of elliptic curves

Torsion modules

• Elements annihilated by non-zero elements of the Dedekind domain
• Finite in the case of modules over rings of integers in number fields
• Correspond to skyscraper sheaves in the geometric picture
• Used in the study of class groups and Selmer groups
• Play a role in the formulation of various duality theorems

Projective modules

• Modules $P$ such that $\text{Hom}(P,-)$ preserves exact sequences
• Finitely generated projective modules over Dedekind domains always free
• Related to vector bundles on the associated one-dimensional scheme
• Used in the study of class groups and K-theory of Dedekind domains
• Play a role in various formulations of class field theory

Dedekind domains and algebraic curves

• Connection between Dedekind domains and algebraic curves fundamental in arithmetic geometry
• Provides a bridge between number theory and algebraic geometry
• Allows for the application of geometric intuition to arithmetic problems

Function fields

• Field of fractions of coordinate ring of a non-singular algebraic curve
• Analogous to number fields in the arithmetic setting
• Dedekind domains arise as rings of regular functions on affine open subsets
• Allow for the development of a geometric theory of divisors and linear systems
• Used in the study of zeta functions and L-functions of curves over finite fields

Divisors on curves

• Formal $\mathbb{Z}$-linear combinations of points on the curve
• Correspond to fractional ideals in the function field setting
• Degree of a divisor analogous to the norm of an ideal in number fields
• Linear equivalence of divisors related to principal ideals
• Riemann-Roch theorem provides a powerful tool for studying divisors on curves

Dedekind zeta functions

• Dedekind zeta functions generalize the Riemann zeta function to number fields
• Provide deep connections between analytic and algebraic aspects of number theory
• Play a crucial role in the study of distribution of prime ideals in number fields

Definition and properties

• Defined as Dirichlet series $\zeta_K(s) = \sum_{I} \frac{1}{N(I)^s}$ where $I$ runs over non-zero ideals
• Admit Euler product factorization over prime ideals
• Have meromorphic continuation to the entire complex plane
• Satisfy a functional equation relating values at $s$ and $1-s$
• Encode important arithmetic information about the number field (class number, regulator)

Relation to ideal class group

• Special values of Dedekind zeta functions related to class number formula
• Residue at $s=1$ involves class number and other arithmetic invariants
• Zeros of Dedekind zeta functions conjectured to have arithmetic significance
• Used in the study of distribution of prime ideals (Prime Ideal Theorem)
• Provide a tool for studying L-functions of motives over number fields

Generalizations of Dedekind domains

• Various generalizations of Dedekind domains exist, extending the concept to broader classes of rings
• These generalizations allow for the study of more complex algebraic and geometric objects
• Understanding these extensions provides insights into the essential properties of Dedekind domains

Krull domains

• Generalize Dedekind domains to higher dimensions
• Characterized by properties of their localizations at prime ideals
• Allow for a theory of divisors in higher-dimensional algebraic geometry
• Include important classes of rings (regular local rings, UFDs)
• Provide a setting for studying factorization in more general contexts

Prufer domains

• Generalize Dedekind domains by relaxing the noetherian condition
• Every finitely generated ideal invertible
• Include important examples (rings of entire functions)
• Allow for a theory of divisibility without unique factorization
• Used in the study of algebraic curves over non-algebraically closed fields

Dedekind-like rings

• Various classes of rings sharing some properties with Dedekind domains
• Include almost Dedekind domains, pseudo-Dedekind domains
• Allow for the study of rings with weaker factorization properties
• Provide insights into which properties of Dedekind domains essential for various results
• Used in investigating limits of arithmetic in more general settings