2024/03/04 by Lázaro O. Rodríguez Díaz, Díaz, Lázaro O. Rodríguez
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2403.02219
openalex publication_date 2024/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by a valuation theorem, recently obtained by Rangachev, we study the étale extensions A⊂ B of polynomial rings over an algebraically closed field of characteristic zero, such that the integral closure A is a primary A-submodule of B. We prove that in this case A has infinite cyclic divisor class group, where the generator is a prime divisor equal to the complement of \textrmSpec(B) in \textrmSpec(A). Moreover, this prime divisor coincides with the ramification divisor of the finite extension A⊂ A. In this situation we carry out Wright's geometric approach for two-dimensional non-integral étale extensions. It follows from the work of Miyanishi that \textrmSpec(A) is a smooth affine surface. We show that \textrmSpec(A) is an \mathbbA1-bundle over ℙ1, more precisely a Danilov-Gizatullin surface of index three. Based on Wright's analysis of which of these affine surfaces can factorize an étale morphism of the complex affine plane and his description of its affine coordinate rings, we prove that under the strong assumption that A is always a primary A-submodule of B, any two-dimensional complex étale extension is integral.