On the integrality of \'etale extensions of polynomial rings

Motivated by a valuation theorem, recently obtained by Rangachev, we study the \'etale extensions $A\subset B$ of polynomial rings over an algebraically closed field of characteristic zero, such that the integral closure $\overline{A}$ is a primary $\overline{A}$-submodule of $B$. We prove that in this case $\overline{A}$ has infinite cyclic divisor class group, where the generator is a prime divisor equal to the complement of $\textrm{Spec}(B)$ in $\textrm{Spec}(\overline{A})$. Moreover, this prime divisor coincides with the ramification divisor of the finite extension $A\subset \overline{A}$. In this situation we carry out Wright's geometric approach for two-dimensional non-integral \'etale extensions. It follows from the work of Miyanishi that $\textrm{Spec}(\overline{A})$ is a smooth affine surface. We show that $\textrm{Spec}(\overline{A})$ is an $\mathbb{A}^{1}$-bundle over $\mathbb{P}^{1}$, more precisely a Danilov-Gizatullin surface of index three. Based on Wright's analysis of which of these affine surfaces can factorize an \'etale morphism of the complex affine plane and his description of its affine coordinate rings, we prove that under the strong assumption that $\overline{A}$ is always a primary $\overline{A}$-submodule of $B$, any two-dimensional complex \'etale extension is integral.