🧮Commutative Algebra Unit 6 – Integral Extensions & Closed Domains

Key Concepts and Definitions

• Integral extension defined as an extension of rings $A \subseteq B$ where every element $b \in B$ is integral over $A$
• Element $b \in B$ is integral over $A$ if it satisfies a monic polynomial with coefficients in $A$
• Closed domain is an integral domain $A$ in which the integral closure of $A$ in its field of fractions is $A$ itself
• Integral closure of a domain $A$ in a field extension $K$ of its field of fractions consists of all elements in $K$ that are integral over $A$
  • Denoted as $\overline{A}$ or $A^{ic}$
• Integrally closed domain is another term for a closed domain
• Monic polynomial has a leading coefficient equal to 1
• Field of fractions of an integral domain $A$ is the smallest field containing $A$, denoted as $\operatorname{Frac}(A)$

Integral Extensions: Basics

• Extension of rings $A \subseteq B$ is integral if every element of $B$ is integral over $A$
• Element $b \in B$ is integral over $A$ if there exists a monic polynomial $f(x) \in A[x]$ such that $f(b) = 0$
  • Polynomial $f(x)$ is called the integral equation for $b$ over $A$
• Integral extensions are always algebraic extensions
  • Algebraic extension means every element in $B$ is a root of some polynomial with coefficients in $A$
• Examples of integral extensions include $\mathbb{Z} \subseteq \mathbb{Z}[i]$ and $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{2})$
• Integral extensions are not necessarily finite extensions
  • Finite extension has a finite degree $[B:A]$, meaning $B$ is a finitely generated $A$-module

Properties of Integral Extensions

• Integral extensions are transitive
  • If $A \subseteq B$ and $B \subseteq C$ are integral extensions, then $A \subseteq C$ is also an integral extension
• Integral extensions are preserved under localization
  • If $A \subseteq B$ is an integral extension and $S$ is a multiplicative subset of $A$, then $S^{-1}A \subseteq S^{-1}B$ is an integral extension
• Integral extensions are preserved under quotients
  • If $A \subseteq B$ is an integral extension and $I$ is an ideal of $A$, then $A/I \subseteq B/IB$ is an integral extension
• Integral extensions are preserved under tensor products
  • If $A \subseteq B$ and $A \subseteq C$ are integral extensions, then $A \subseteq B \otimes_A C$ is an integral extension
• Integral extensions are preserved under finite direct sums
  • If $A \subseteq B_i$ are integral extensions for $i = 1, \ldots, n$, then $A \subseteq \bigoplus_{i=1}^n B_i$ is an integral extension

Closed Domains: Fundamentals

• Closed domain is an integral domain $A$ in which the integral closure of $A$ in its field of fractions is $A$ itself
  • Equivalently, $A$ is a closed domain if $A = \overline{A}$
• Examples of closed domains include fields, unique factorization domains (UFDs), and Dedekind domains
• Closed domains are integrally closed in any ring extension
  • If $A$ is a closed domain and $A \subseteq B$ is a ring extension, then $A$ is integrally closed in $B$
• Integral closure of a domain $A$ in a field extension $K$ of its field of fractions is the intersection of all valuation rings of $K$ containing $A$
• Valuation ring is an integral domain with a unique maximal ideal
• Closed domains satisfy the ascending chain condition (ACC) on principal ideals
  • Every ascending chain of principal ideals in a closed domain stabilizes

Relationships Between Integral Extensions and Closed Domains

• If $A \subseteq B$ is an integral extension and $A$ is a closed domain, then $B$ is also a closed domain
• If $A \subseteq B$ is an integral extension and $B$ is a closed domain, then $A$ is not necessarily a closed domain
  • Counterexample: $\mathbb{Z} \subseteq \mathbb{Z}[i]$ is an integral extension, $\mathbb{Z}[i]$ is a closed domain (UFD), but $\mathbb{Z}$ is not a closed domain
• If $A \subseteq B$ is an integral extension and $B$ is a finitely generated $A$-module, then $A$ is a closed domain if and only if $B$ is a closed domain
• If $A \subseteq B$ is an integral extension and $A$ is a Noetherian domain, then $A$ is a closed domain if and only if $B$ is a closed domain
• Integral closure of a Noetherian domain in a finite extension of its field of fractions is a finite $A$-module

Theorems and Proofs

• Going-Up Theorem: If $A \subseteq B$ is an integral extension and $P \subseteq Q$ are prime ideals of $A$, then there exist prime ideals $P' \subseteq Q'$ of $B$ such that $P' \cap A = P$ and $Q' \cap A = Q$
  • Proof involves using Zorn's Lemma and properties of integral extensions
• Going-Down Theorem: If $A \subseteq B$ is an integral extension, $A$ is an integrally closed domain, and $Q' \subseteq P'$ are prime ideals of $B$, then $Q' \cap A \subseteq P' \cap A$
  • Proof relies on the Going-Up Theorem and properties of integral extensions
• Lying-Over Theorem: If $A \subseteq B$ is an integral extension and $P$ is a prime ideal of $A$, then there exists a prime ideal $Q$ of $B$ such that $Q \cap A = P$
  • Special case of the Going-Up Theorem
• Cohen-Seidenberg Theorems: A collection of theorems relating prime ideals in integral extensions, including the Going-Up, Going-Down, and Lying-Over Theorems
• Krull-Akizuki Theorem: If $A$ is a one-dimensional Noetherian domain and $A \subseteq B$ is an integral extension, then $B$ is a Noetherian domain

Applications in Algebraic Geometry

• Integral extensions and closed domains play a crucial role in the study of algebraic varieties and schemes
• Affine varieties are defined as the zero sets of polynomials in affine space over an algebraically closed field
  • Coordinate rings of affine varieties are finitely generated $k$-algebras that are integral domains
• Projective varieties are defined as the zero sets of homogeneous polynomials in projective space over an algebraically closed field
  • Homogeneous coordinate rings of projective varieties are graded, finitely generated $k$-algebras that are integral domains
• Normalization of an affine or projective variety corresponds to taking the integral closure of its coordinate ring in its field of fractions
  • Normalized varieties have coordinate rings that are integrally closed domains
• Schemes are a generalization of algebraic varieties that allow for more general base rings and local structure
  • Integral extensions and closed domains are essential in the construction and study of schemes

Problem-Solving Techniques

• To determine if an extension $A \subseteq B$ is integral, find monic polynomials with coefficients in $A$ for each element of $B$
• To check if a domain $A$ is a closed domain, compute its integral closure $\overline{A}$ in its field of fractions and verify that $A = \overline{A}$
  • Integral closure can be computed by finding the intersection of all valuation rings containing $A$
• To prove statements about integral extensions and closed domains, use properties such as transitivity, preservation under localization, quotients, tensor products, and finite direct sums
• Apply the Going-Up, Going-Down, and Lying-Over Theorems to relate prime ideals in integral extensions
• Use the Krull-Akizuki Theorem to deduce properties of integral extensions of Noetherian domains
• In algebraic geometry, consider normalization techniques to study varieties with integrally closed coordinate rings
• Utilize the connection between integral extensions, closed domains, and the structure of schemes in algebraic geometry