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

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 $$A$$?

• When is local henselian (e.g., local artinian), then the finite etale A have essentially the same description as in the field case (but there are also nonfinite etale A that don't extend to finite B). If R is a normal domain, you essentially get them all by taking the integral closure A in a finite separable extension of the field of fractions of R, and passing to an open subset of Spec(A). It is true that locally on Spec(A), the extension is given by a single polynomial, but this doesn't tell you much globally. For a discussion of such things, see, for example, Chapter I of Milne's book EtCo.
– anon
Apr 26, 2019 at 12:59

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.