6
$\begingroup$

Let $S$ be a smooth variety over $\mathbb C$ or a smooth quasi-projective integral scheme over Spec $\mathbb{Z}$.

Let $G$ be an (abstract) discrete group. For instance, $G =\mathbb{Z}^n$ or $G$ a finite index torsionfree subgroup of $SL_2(\mathbb{Z})$. I want to exclude finite groups, but also infinite products of finite groups. (Actually, I am only interested in non-abelian torsionfree Fuchsian groups.)

Is every $G$-torsor over $S$ trivial for the etale topology on $S$? If not, what are some interesting examples?

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Am I right in assuming that such things are classified by (continuous?) homomorphisms from the etale fundamental group of $S$ to $G$? If so, there should be plenty of examples as soon as $G$ has finite subgroups, which for example $SL_2(\mathbb{Z})$ has several of. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 23, 2017 at 21:35
  • $\begingroup$ Good point. I edited the question. $\endgroup$
    – Sam
    Commented Sep 23, 2017 at 22:10
  • $\begingroup$ The OP does specify that $G$ should be torsionfree. I think the OP wants somebody to say, a torsor over a regular base for the 'etale topology necessarily factors through a finite subgroup of the structure group. $\endgroup$
    – Jason Starr
    Commented Sep 23, 2017 at 22:11
  • $\begingroup$ How do you define torsor? $\endgroup$
    – Roman Fedorov
    Commented Sep 25, 2017 at 17:47

0

Reset to default

You must log in to answer this question.