Let $\mathcal{O}$ be an order in a number field $K$, that is a subring of $K$ with rank as abelian group equal to $[K:\mathbb{Q}]$. What is known about the SGA3-étale fundamental group of $X=\mathrm{Spec}(\mathcal{O})$ ? Are there example where it is not profinite ? My motivation for asking this question is for computing the étale cohomology group $H^1(X,\mathbb{Z})=\mathrm{Hom}_{cont}(\pi_1^{SGA3}(X),\mathbb{Z})$.

If X is geometrically unibranch then the SGA3 étale fundamental group equals the étale fundamental group, hence is profinite and the cohomology group vanishes.

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    $\begingroup$ Sure, you can create a nodal nodal order by gluing points together, just the same the way you produce a nodal curve. The ring of functions on the nodal affine line is $\{f\in \mathbb k[x]\;|\;f(0)=f(1)\}$. Similarly, take the order $\{x\in\mathbb Z[i]\;|\;x+(1-2i)\equiv x+(1+2i)\in \mathbb Z/5\}$ using the identifications $\mathbb Z[i]/(1+2i)=\mathbb Z/5=\mathbb Z[i]/(1-2i)$. $\endgroup$ Oct 8, 2020 at 0:49

1 Answer 1


I only vaguely know about étale fundamental groups at the moment so I went the other way and computed $H^1(X,\mathbb{Z})$. Let $\pi : Y=\mathrm{Spec}(\mathcal{O}_K) \to X$ denote the normalization. Denote $Z$ the singular locus and put $s_v=\#\pi^{-1}(v)-1$ for every $v\in Z$.

Claim : $H^1(X,\mathbb{Z})$ is a free abelian group of rank $\sum_{v\in Z}s_v$.

Proof : Consider the cokernel $F$ of the injection $0\to \mathbb{Z} \to \pi_\ast \mathbb{Z}$. Then $F$ is a skyscraper sheaf supported on singular points and we can compute by Galois cohomology that $H^0(X,F)\simeq \oplus_{v\in Z}\mathbb{Z}^{s_v}$ (this comes from the vanishing of the $H^1$ in Galois cohomology with coefficients in $\mathbb{Z}$). Because $H^1(Y,\mathbb{Z})=0$ since $Y$ is normal (see etale-cohomology-with-coefficients-in-the-integers), the long exact sequence in cohomology reads $$ 0 \to \mathbb{Z} \xrightarrow{\mathrm{Id}} \mathbb{Z} \to H^0(X,F) \to H^1(X,\mathbb{Z}) \to 0 $$ which proves the claim. $\blacksquare$

In particular, as soon as there is a singular point with at least two primes above it in the normalization, the (SGA3) étale fundamental group of $X$ be not profinite.


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