[Paper Review] An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case

Kazuhiko Kurano, Kazuma Shimomoto|arXiv (Cornell University)|Oct 13, 2015

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper presents a new, elementary proof of the Cohen-Gabber theorem in the equal characteristic $p>0$ case, establishing the existence of a coefficient field and a system of parameters such that the associated power series ring embeds module-finitely into the complete local ring, with generically separable field extensions to quotient fields of dimension $d$. The proof uses induction on the number of extra generators of the maximal ideal and Weierstrass preparation to construct the desired coefficient field and parameters.

ABSTRACT

The aim of this article is to give a new proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case.

Motivation & Objective

• To provide a new, elementary proof of the Cohen-Gabber theorem in the equal characteristic $p>0$ case.
• To construct a coefficient field $\phi: k \to A$ and a system of parameters $y_1,\dots,y_d$ such that $\phi(k)[[y_1,\dots,y_d]] \subset A$ is module-finite.
• To ensure that the field extension $\operatorname{Frac}(\phi(k)[[y_1,\dots,y_d]]) \to \operatorname{Frac}(A/P)$ is separable for any minimal prime $P$ with $\dim A/P = d$.
• To generalize the classical Cohen structure theorem by achieving generic étale behavior in the reduced equi-dimensional case.

Proposed method

• Use induction on $h$, the number of extra generators beyond a system of parameters in the maximal ideal of $A$
• Apply the Weierstrass Preparation Theorem to factor power series in $A[[X]]$ into a unit and a distinguished polynomial
• Construct a coefficient field $\phi: k \to A$ using a $p$-basis of the residue field $k$ over $\mathbb{F}_p$
• Build a chain of module-finite extensions through intermediate rings $B$, $C$, $D$, and $E$ in a commutative diagram
• Verify separability of field extensions by analyzing the behavior of differentials and the structure of quotient fields
• Use the fact that separability is preserved under base change and adjoining elements in the module-finite setting

Experimental results

Research questions

• RQ1 Can the Cohen-Gabber theorem in equal characteristic $p>0$ be proven using only elementary methods from commutative algebra?
• RQ2 Is it possible to construct a coefficient field and system of parameters such that the associated power series ring embeds module-finitely into $A$ with separable residue field extensions?
• RQ3 How can the generic étale property of the extension be achieved in the reduced equi-dimensional case?
• RQ4 What role does the Weierstrass Preparation Theorem play in constructing such coefficient fields and parameters?

Key findings

• The paper establishes the existence of a coefficient field $\phi: k \to A$ such that $\pi \circ \phi = \mathrm{id}_k$, where $\pi: A \to k$ is the quotient map.
• A system of parameters $y_1, \dots, y_d$ is constructed such that $\phi(k)[[y_1, \dots, y_d]] \subset A$ is a module-finite extension.
• For any minimal prime $P \subset A$ with $\dim A/P = d$, the field extension $\operatorname{Frac}(\phi(k)[[y_1, \dots, y_d]]) \to \operatorname{Frac}(A/P)$ is separable.
• The proof uses induction on $h$, the number of generators of $\mathfrak{m}$ beyond a system of parameters, reducing the problem to the $h=1$ case.
• The construction ensures that all intermediate maps in the diagram are injective and module-finite, preserving the required properties.
• An example is provided showing that separability fails over certain coefficient fields but can be achieved by choosing a suitable coefficient field, such as $\mathbb{F}_p(s)$ with $s = t + X$.