# If $R$ is an etale extension of $\mathbb Z$, then $R = \mathbb Z^n$?

Related question.

Let $A$ be a ring, and let $B$ be an $A$-algebra which is projective and finite as an $A$-module. Then the trace $B \rightarrow A$ can be defined. Let's say that $B$ is separable over $A$ if the trace induces an isomorphism of $A$-modules $B \rightarrow \operatorname{Hom}_A(B,A)$. For the purposes of this question, let's say that $B$ is an etale $A$-algebra if it is finite, projective, and separable.

Is it true that every etale $\mathbb Z$-algebra is of the form $\mathbb Z^n$? This is stated in these notes, example 1.1.13(ii).

It does not seem so easy to prove. If this is true, then combined with the much easier result, "if $L/K$ is an unramified extension of number fields, then $\mathcal O_L$ is an etale $\mathcal O_K$-algebra," we would obtain an algebraic proof that there are no nontrivial unramified extensions of $\mathbb Q$. I had thought that the only known proofs of this result relied on Minkowski's bound for the discriminant. Perhaps any proof that etale $\mathbb Z$-algebras are of the form $\mathbb Z^n$ must rely on this fact.

• This fact is equivalent to Minkowski's theorem. I do not know of a different proof. – Keerthi Madapusi Pera Jul 23 '18 at 18:38
• A better term would be finite étale $A$-algebra, because there are also étale $A$-algebras that are not finite. – R. van Dobben de Bruyn Jul 23 '18 at 19:59
• @R.vanDobbendeBruyn Do you know where to read about non-finite étale $\mathbb Z$-algebras? – მამუკა ჯიბლაძე Jul 24 '18 at 4:57
• @მამუკაჯიბლაძე: I don't know a reference that treats the affine case exclusively; only the more general setting of étale morphisms of schemes. Examples include $\operatorname{Spec} \mathbb Z[1/n]$ and finite étale extensions thereof (e.g. $\mathbb Z[\zeta_n][1/n]$, the integral closure of $\mathbb Z[1/n]$ in the ray class field of modulus $n \cup \infty$), as well as products of such. – R. van Dobben de Bruyn Jul 24 '18 at 13:47
• @მამუკაჯიბლაძე: admittedly, there might not be a reference that I truly like. Hartshorne leaves it at four exercises, and in EGA and the Stacks project it relies on a lot of theory already. A place to start could be Milne's notes on étale cohomology, or his book on the same subject (which is more technical). – R. van Dobben de Bruyn Jul 24 '18 at 14:12

Let $X$ be a normal integral scheme. Then $\pi_1^\mathrm{et}(X)$ is isomorphic to $\mathrm{Gal}(K(X)^\mathrm{un}/K(X))$, where $K(X)^\mathrm{un}$ is the compositum of all finite extensions $F$ of $K(X)$ (in a fixed algebraic closure of $K(X)$) for which $X$ is unramified in $F$. Applying this to $X = \mathrm{Spec}(\mathbf{Z})$, it follows the result you are asking about is equivalent to Minkowski's theorem (that there are no nontrivial unramified extensions of $\mathbf{Q}$).
• By the way, the vanishing of $\pi_1^\mathrm{et}(\mathrm{Spec}(\mathbf{Z}))$ is just (by class field theory) the statement that $\mathbf{Z}$ is a PID. – skd Jul 24 '18 at 2:02