Standard étale algebra

Let $A$ be a commutative ring. An $A$-algebra $B$ is standard étale if there is a polynomial $P\in A[T]$ with $P$ monic and an element $g\in A[T]$ such that $B\cong (A[T]\mathrm{/}(P){)}_g$ as $A$-algebras and the image of the formal derivative $P^{\mathrm{\prime }}$ is a unit of $B$. Equivalently (with $P$ monic and the isomorphism fixed), $P^{\mathrm{\prime }}$ has invertible image in $(A[T]\mathrm{/}(P){)}_g$ if and only if $g^n\in (P^{\mathrm{\prime }})$ for some $n\ge 0$, where $(P^{\mathrm{\prime }})$ denotes the ideal of $A[T]\mathrm{/}(P)$ generated by the image of $P^{\mathrm{\prime }}$; this is the form in which the condition is usually verified.

Two conventions belong to the definition. First, $P$ is required to be monic, so $A[T]\mathrm{/}(P)$ is a free $A$-module with basis $1,T,\dots ,T^{\mathrm{deg}P-1}$ by division with remainder, before any localisation is performed; the localised algebra $(A[T]\mathrm{/}(P){)}_g$ itself need not be finite over $A$. Second, the presentation is part of the data: the same $A$-algebra can be standard étale through one presentation and not visibly standard étale through another, and the condition is required to hold for at least one monic presentation with unit derivative. A standard étale algebra is étale over $A$ in the sense of Étale morphism of schemes when $A\to B$ is finitely presented, which it is here because $A[T]\mathrm{/}(P)$ is a finitely presented $A$-algebra and localisation preserves finite presentation; the derivative condition is the Jacobian condition of the single equation $P$ computed by Differentials of a polynomial quotient and the Jacobian cokernel.