Structure theorem for etale algebras over a more general ring than a field

Question

I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition). In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite product of finite separable extensions of $k$.

What about algebras over more general rings? Do we get a decomposition of $A$ as a finite product of "simpler" rings?

For example, what if:

$R$ is local artinian

$R$ is artinian

$R$ is local noetherian of dimension 1

$R$ local noetherian domain of dimension 1

$R$ is a DVR

$R$ is a PID

$R$ is a Dedekind domain

and so on...

I would be interested in any of these cases and in others you may want to add ( I hope the question is not too broad)

In cases where $R$ is normal, can we use Serre's criterion to get a decomposition of $A$?

Answer 1

I would recommend you to look at EGA IV$_4$ (18.4.5) and (18.4.6). In brief: Let $A \to B$ be a locally finite presentation algebra with $A$ local with maximal $\mathfrak{m}$. Then $B$ is étale at a point corresponding to a maximal $\mathfrak{n}$ If and only if there is a polynomial $F \in A[T]$ such that $B = A[T]/\langle F \rangle$ such that $F'(\overline{T}) \notin \mathfrak{n}$. We sometimes say that $F$ is separable at $\mathfrak{n}$. Every other case restricts essentially to the local case by localizing $B$ adequately.