Strict Henselization Of A Point In Affine Spaces A Comprehensive Guide

Introduction

In the realm of algebraic geometry and commutative algebra, understanding the local structure of schemes is paramount. The strict henselization is a powerful tool for probing the local properties of algebraic varieties, particularly at a given point. This article delves into the specific case of the strict henselization of a point in affine space over an algebraically closed field. Our primary focus is to elucidate the structure of the strict henselian trait $S$ of the origin $0$ in $\mathbb{A}^n$, and, crucially, to identify its generic point. We will explore the concept of henselian rings and their relevance in studying local properties of schemes, and we will unravel the construction of the strict henselization in the context of affine spaces. The significance of this investigation lies in its applications to étale cohomology and other advanced topics in algebraic geometry, where the local structure of schemes plays a vital role. By characterizing the strict henselization and its generic point, we gain deeper insights into the local behavior of algebraic varieties and their associated algebraic structures. This exploration is not merely an academic exercise; it provides the foundational understanding necessary for tackling more intricate problems in modern algebraic geometry and related fields.

Background on Henselian Rings

To properly understand the strict henselization, we must first lay the groundwork by discussing henselian rings. A henselian ring is a local ring that satisfies a certain lifting property for polynomial equations. More formally, let $(A, \mathfrak{m})$ be a local ring, where $A$ is the ring and $\mathfrak{m}$ is its maximal ideal. We say that $A$ is henselian if Hensel's lemma holds for $A$. Hensel's lemma, in this context, is a powerful statement about lifting roots of polynomials. It states that if we have a monic polynomial $f(x) \in A[x]$ and a root $\overline{a}$ of the reduction of $f(x)$ modulo $\mathfrak{m}$ such that the derivative of $f$ evaluated at $\overline{a}$ is a unit in $A/\mathfrak{m}$, then there exists a root $a \in A$ of $f(x)$ that lifts $\overline{a}$. This property allows us to solve polynomial equations locally in a henselian ring, and it is a crucial tool in studying the structure of local rings.

Henselian rings generalize the notion of complete local rings. Recall that a local ring is complete if it is complete with respect to its maximal ideal. While every complete local ring is henselian, the converse is not necessarily true. Henselian rings offer a weaker notion of completeness that is still sufficient for many applications in algebraic geometry. The strict henselization, which we will discuss later, provides a way to henselize a local ring, making it henselian while preserving certain properties. The notion of henselian rings is central to understanding the local behavior of schemes. It allows us to study the solutions of polynomial equations in a local setting and to analyze the structure of rings and modules over local rings. The henselian property is particularly important in the context of étale morphisms, which are smooth morphisms that locally look like isomorphisms. Henselian rings provide a convenient setting for studying étale morphisms and their properties. Understanding henselian rings is a crucial step towards grasping the more advanced concepts in algebraic geometry, such as étale cohomology and the fundamental group of a scheme. The henselian property allows us to work with local information in a more controlled manner, making it a valuable tool in the study of algebraic varieties and their properties.

Strict Henselization: Definition and Construction

The strict henselization of a local ring is a specific henselian ring that captures the étale-local properties of the original ring. Let $(A, \mathfrak{m})$ be a local ring. The strict henselization of $A$, denoted by $A^{sh}$, is the direct limit of all étale neighborhoods of $A$. An étale neighborhood of $A$ is an étale $A$-algebra $B$ such that the map $\operatorname{Spec}(B) \to \operatorname{Spec}(A)$ induces an isomorphism of residue fields. In other words, $B/\mathfrak{m}_B \cong A/\mathfrak{m}$, where $\mathfrak{m}_B$ is a maximal ideal of $B$ lying over $\mathfrak{m}$.

More concretely, we can construct the strict henselization as follows: Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and residue field $k = A/\mathfrak{m}$. Let $k^{sep}$ be a separable closure of $k$. The strict henselization $A^{sh}$ of $A$ is the henselian local ring obtained by taking the direct limit of all étale $A$-algebras $B$ such that $B/\mathfrak{m}_B$ is a subfield of $k^{sep}$ containing $k$. This construction ensures that $A^{sh}$ is a henselian ring with residue field $k^{sep}$. The strict henselization is unique up to isomorphism and enjoys several important properties. First, it is a henselian ring, meaning that Hensel's lemma holds for $A^{sh}$. Second, its residue field is a separable closure of the residue field of $A$. Third, it is a universal object in the sense that any henselian local ring that receives a local homomorphism from $A$ also receives a local homomorphism from $A^{sh}$. The strict henselization plays a central role in étale cohomology and the study of the étale fundamental group. It provides a way to replace a local ring with a henselian ring that has a separably closed residue field, which is often necessary for applying étale cohomology techniques. Understanding the strict henselization is crucial for working with étale morphisms and their properties, and it is a fundamental tool in modern algebraic geometry.

The Case of Affine Space

Now, let us specialize to the case where we are working over an algebraically closed field $k$ and consider affine space $\mathbb{A}^n$ over $k$. We are interested in the strict henselization of the local ring of $\mathbb{A}^n$ at the origin, which we denote by $0$. The coordinate ring of $\mathbb{A}^n$ is $k[x_1, \dots, x_n]$, and the local ring at the origin is obtained by localizing this ring at the maximal ideal $\mathfrak{m} = (x_1, \dots, x_n)$. Thus, we have

$$A = k[x_1, \dots, x_n]_{(x_1, \dots, x_n)}.$$

Since $k$ is algebraically closed, the residue field of $A$ is $A/\mathfrak{m} = k$, which is already separably closed. Therefore, the strict henselization $A^{sh}$ in this case is obtained by taking the direct limit of all étale neighborhoods of $A$. An étale neighborhood of $A$ is an étale $A$-algebra $B$ such that $B/\mathfrak{m}_B \cong A/\mathfrak{m} = k$. In this particular situation, the strict henselization has a relatively simple description. It turns out that the strict henselization of $A$ is isomorphic to the henselization of $A$. The henselization of $A$ is obtained by completing $A$ with respect to the $\mathfrak{m}$-adic topology and then taking the algebraic closure of the quotient field of the completion. In our case, the completion of $A$ with respect to the $\mathfrak{m}$-adic topology is the ring of formal power series $k[[x_1, \dots, x_n]]$. The quotient field of this ring is the field of fractions of formal power series, and the algebraic closure of this field is a large and complicated object. However, the henselization of $A$ is a much simpler object. It is the direct limit of all étale $A$-algebras, and it can be thought of as a kind of algebraic closure of $A$ in a suitable sense. The strict henselization of $A$ is the same as the henselization of $A$ in this case because the residue field $k$ is algebraically closed. This simplifies the situation considerably, and it allows us to understand the structure of the strict henselization more explicitly. The strict henselization of the local ring of affine space at a point is a fundamental object in algebraic geometry, and understanding its structure is crucial for studying the local properties of algebraic varieties.

The Strict Henselian Trait

The strict henselian trait $S$ of $0 \in \mathbb{A}^n$ is the spectrum of the strict henselization of the local ring of $\mathbb{A}^n$ at $0$. In other words, if $A = k[x_1, \dots, x_n]_{(x_1, \dots, x_n)}$ is the local ring of $\mathbb{A}^n$ at the origin, then $S = \operatorname{Spec}(A^{sh})$, where $A^{sh}$ is the strict henselization of $A$. The strict henselian trait is a local scheme that captures the étale-local properties of $\mathbb{A}^n$ at the origin. It is a powerful tool for studying the local behavior of algebraic varieties and for proving theorems about étale morphisms and étale cohomology. The strict henselian trait has two points: the closed point and the generic point. The closed point corresponds to the maximal ideal of $A^{sh}$, which is the unique maximal ideal since $A^{sh}$ is a local ring. The generic point corresponds to the zero ideal of $A^{sh}$, which is the prime ideal consisting of all nilpotent elements. The closed point of $S$ is the image of the origin $0 \in \mathbb{A}^n$ under the canonical morphism $\operatorname{Spec}(A^{sh}) \to \operatorname{Spec}(A)$. It represents the local properties of $\mathbb{A}^n$ at the origin. The generic point of $S$, on the other hand, is more subtle. It represents the generic properties of the strict henselization, and it is closely related to the étale fundamental group of $\mathbb{A}^n$ at the origin. Understanding the generic point of the strict henselian trait is crucial for studying the étale cohomology of $\mathbb{A}^n$ and for proving theorems about étale coverings. The strict henselian trait is a fundamental object in algebraic geometry, and its structure and properties are essential for understanding the local behavior of algebraic varieties and their étale topology.

Identifying the Generic Point of $S$

The crucial question is: What is the generic point of $S$? The generic point of $S$ corresponds to the minimal prime ideal of $A^{sh}$. Since $A^{sh}$ is a henselian ring, it is an integral domain. This is a key property of strict henselizations. Because $A^{sh}$ is an integral domain, its minimal prime ideal is simply the zero ideal $(0)$. Therefore, the generic point of $S$ corresponds to the zero ideal in $A^{sh}$. This implies that the generic point of $S$ is the prime ideal $(0) \subset A^{sh}$. In more geometric terms, the generic point of $S$ represents the