🧮Commutative Algebra Unit 11 – Completion and Henselian Rings

Key Concepts and Definitions

• Completion is a process that extends a ring or module with respect to a given ideal or filtration
• Henselian rings are local rings that satisfy Hensel's lemma, a lifting property for certain polynomial equations
• Krull topology on a ring $R$ is defined by a filtration of ideals $I_1 \supset I_2 \supset \cdots$
  • The completion of $R$ with respect to this topology is denoted as $\hat{R}$
• Jacobson radical of a ring $R$, denoted by $J(R)$, is the intersection of all maximal ideals of $R$
• Nilradical of a ring $R$, denoted by $nil(R)$, is the ideal consisting of all nilpotent elements of $R$
• Étale morphism is a smooth morphism of relative dimension zero between schemes
• Henselization of a local ring $(R,\mathfrak{m})$ is the smallest Henselian ring between $R$ and its completion $\hat{R}$

Historical Context and Motivation

• The concept of completion originated in the work of Hensel on $p$-adic numbers in the early 20th century
• Hensel introduced the notion of $p$-adic numbers as completions of the rational numbers with respect to the $p$-adic valuation
• The motivation behind completion was to extend the field of rational numbers to include limits of Cauchy sequences
  • This construction allowed for the development of $p$-adic analysis and $p$-adic geometry
• Henselian rings were introduced by Azumaya in the 1950s as a generalization of complete local rings
• The study of Henselian rings is motivated by their applications in algebraic geometry and number theory
  • Henselian rings provide a framework for studying local properties of schemes and algebraic varieties

Properties of Completion

• The completion $\hat{R}$ of a ring $R$ with respect to an ideal $I$ is a complete Hausdorff topological ring
• There exists a canonical ring homomorphism $\phi: R \to \hat{R}$, which is an isomorphism if and only if $R$ is complete with respect to the $I$-adic topology
• If $R$ is Noetherian and $I$ is finitely generated, then $\hat{R}$ is also Noetherian
• Krull Intersection Theorem: If $R$ is Noetherian and $I$ is proper, then $\bigcap_{n=1}^\infty I^n = 0$ in $\hat{R}$
• For a Noetherian local ring $(R,\mathfrak{m})$, the completion $\hat{R}$ with respect to $\mathfrak{m}$ is faithfully flat over $R$
• If $R$ is a finitely generated algebra over a complete local ring, then $R$ is a quotient of a power series ring

Henselian Rings: Basics and Examples

• A local ring $(R,\mathfrak{m})$ is Henselian if Hensel's lemma holds: for any monic polynomial $f(x) \in R[x]$ and any simple root $\bar{a} \in R/\mathfrak{m}$ of $\bar{f}(x)$, there exists $a \in R$ such that $f(a) = 0$ and $a \equiv \bar{a} \pmod{\mathfrak{m}}$
• Examples of Henselian rings include complete local rings, convergent power series rings, and rings of algebraic integers
• Every complete local ring is Henselian, but the converse is not true in general
  • For example, the ring of algebraic integers is Henselian but not complete
• Henselian rings are closed under finite direct products and quotients by ideals
• If $(R,\mathfrak{m})$ is Henselian and $I$ is an ideal of $R$, then $(R/I, \mathfrak{m}/I)$ is also Henselian
• Henselization is a functorial construction that assigns to each local ring its smallest Henselian extension

Comparison: Completion vs. Henselization

• For a local ring $(R,\mathfrak{m})$, its Henselization $R^h$ is always contained in its completion $\hat{R}$
  • The inclusion $R^h \hookrightarrow \hat{R}$ is an equality if and only if $\hat{R}$ is separable over $R$
• Henselization preserves étale morphisms, while completion does not in general
  • If $f: A \to B$ is an étale morphism of local rings, then $f^h: A^h \to B^h$ is also étale
• Completion and Henselization coincide for excellent rings, which include complete Noetherian local rings and finitely generated algebras over fields
• Henselization can be viewed as a "algebraic" approximation of completion, preserving certain algebraic properties while not necessarily being complete

Applications in Number Theory

• Completions and Henselizations play a crucial role in local class field theory
  • For a local field $K$, its completion $\hat{K}$ and Henselization $K^h$ are used to define the local Artin map and study the Galois group of $\hat{K}$ over $K$
• Hensel's lemma is used to construct $p$-adic $L$-functions, which are analogues of complex $L$-functions in the $p$-adic setting
• Henselian rings are used in the study of local-global principles, such as the Hasse principle for quadratic forms and the Grunwald-Wang theorem
• The completions of rings of integers in number fields are used to define $p$-adic valuations and study $p$-adic Galois representations
• Henselian rings are used in the construction of the étale fundamental group of a scheme, which is a key object in étale cohomology

Advanced Theorems and Proofs

• Cohen Structure Theorem: If $R$ is a complete Noetherian local ring with residue field $k$, then there exists a surjective ring homomorphism $W(k)[[x_1, \ldots, x_n]] \to R$, where $W(k)$ is the ring of Witt vectors over $k$
• Nagata's Theorem: If $R$ is a Henselian local ring, then every finitely generated $R$-algebra is a finite $R$-module
• Artin Approximation Theorem: If $R$ is a Henselian local ring and $f_1, \ldots, f_n \in R[[x_1, \ldots, x_m]]$ are formal power series, then any solution to the system $f_1 = \cdots = f_n = 0$ in the completion $\hat{R}$ can be approximated by a solution in $R$
• Elkik's Theorem: If $R$ is a Henselian local ring and $f: X \to \operatorname{Spec}(R)$ is a proper morphism of finite presentation, then $f$ has a section if and only if its base change to the completion $\hat{R}$ has a section
• Popescu's Theorem: Every regular Noetherian ring is a filtered colimit of smooth algebras over $\mathbb{Z}$

Exercises and Problem-Solving Strategies

• When working with completions, it is often helpful to consider the associated graded ring $\operatorname{gr}_I(R) = \bigoplus_{n=0}^\infty I^n/I^{n+1}$ with respect to an ideal $I$
• To prove that a ring is Henselian, try to verify Hensel's lemma directly or show that the ring is a filtered colimit of Henselian rings
• When dealing with étale morphisms and Henselian rings, consider using the functorial properties of Henselization and the fact that Henselization preserves étale morphisms
• In problems involving both completion and Henselization, compare the properties of $R$, $\hat{R}$, and $R^h$, and use the relationship between these rings
• When working with applications in number theory, consider the specific properties of the rings involved, such as rings of integers in number fields or $p$-adic fields
• In proofs involving advanced theorems, break down the problem into smaller steps and use the hypotheses of the theorems to guide your reasoning
• Practice solving problems from various sources, such as textbooks, research papers, and qualifying exams, to develop a deep understanding of the concepts and techniques involved in the study of completions and Henselian rings