Alternative definitions of étale and formally unramified in Wraith

Question

I have stumbled upon the following definitions in a paper by Gavin Wraith.

Definition 1. Say a ring morphism $A\overset \varphi \to B$ is formally unramified if every $b\in B$ admits:

• $b_0\in B$ which is a simple root of some $\varphi (p)$ for some monic $p\in A[x]$;
• monics $f,g\in A[x]$ such that $\varphi (g)(b_0)\in B^\times $ and $b=\tfrac{\varphi(f)(b_0)}{\varphi(g)(b_0)}$.

Definition 2. Say a ring morphism $A\overset \varphi \to B$ is étale if there exist comaximal (generating the unit ideal) elements $b_1,\dots ,b_r\in B$ such that for each $1\leq i\leq r$ there's a $B$ algebra isomorphism $$B[\tfrac 1{b_i}]\cong \tfrac{A[x_1,\dots,x_n][1/g]}{\langle f_1,\dots ,f_n\rangle},\;\det\tfrac{\partial f_i}{\partial x_j}\mid g.$$

Question. When are these definitions equivalent to the modern ones involving lifts and square-zero ideals? (And how to prove the equivalence?)

Answer 1

Following a suggestion made in the comments, I post this as an answer.

These are the "local structure theorems" of unramified and étale algebras. They are equivalent to the definitions with liftings and square-zero ideals. The proofs of the equivalence are not straightforward as far as I know: they are elementary but intricate and a bit lengthy. An excellent reference for all this is Michel Raynaud, Anneaux locaux henseliens; more precisely, see I. Prop. 8, V. Theorem 1 and Theorem 5.