14
$\begingroup$

On a course that ended some time ago, I was handed the following problem:

Problem: Compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}, \overline x)$.

Hint: Find all finite algebraic extensions of $\mathbb{C}((t))$.

It's easy to compute the absolute Galois group of $\mathbb{C}((t))$. Also, I know there are at least two different ways to compute $\pi_1^{ét} (\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \})$:

  • by using the comparison with the profinite completion of the topological fundamental group,

  • by using Hurwitz formula (cf. here).

However, I wonder, if it is possible to formally finish this proof using the above hint.

Question: is it possible to show directly that $\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}$ and $\DeclareMathOperator{\Spec}{Spec} \Spec \mathbb{C}((t))$ have the same étale fundamental group?

The intuition is clear: $\DeclareMathOperator{\Spec}{Spec} \Spec \mathbb{C}((t)) = \Spec \mathbb C[[t]] \setminus \{ (t) \}$ is a "formal" punctured disc and thus should "morally" have the same homotopy type as $\mathbb{A}^1_{\mathbb C} \setminus \{ 0 \}$. But is it possible to formalize this intuition (possibly even using some higher machinery from étale homotopy theory)? Does it generalize to higher dimensions?

It seems to me that the answer would be quite informative, not only for this problem.

I am asking this question here, since a similar question posted on Stack Exchange did not get any satisfactory (from my point view) answer.

$\endgroup$
  • 1
    $\begingroup$ +1, especially for the subquestion "Does it generalize to higher dimensions?", which I interpret as asking about the etale homotopy type of $\mathbb{A}^n \setminus \{0\}$ versus its formal completion near zero. $\endgroup$ – Theo Johnson-Freyd Jun 25 '19 at 0:39
  • 1
    $\begingroup$ I'm not an algebraic geometer. I think the hint was intended as: an etale cover of $\mathbb{A}^1 \setminus \{0\}$ restricts to an etale cover of $\mathrm{Spec}(\mathbb{C}((t))) = \mathbb{D}^1 \setminus \{0\}$; now classify those, and show that each one extends uniquely to $\mathbb{A}^1 \setminus \{0\}$. But this is a disappointing interpretation: as you point out, it is more satisfying to have a general criterion that says that the inclusion $\mathbb{D}^k \setminus\{0\} \subset \mathbb{A}^k \setminus\{0\}$ is an etale homotopy equivalence. $\endgroup$ – Theo Johnson-Freyd Jun 25 '19 at 0:42
  • $\begingroup$ @TheoJohnson-Freyd: I thought of using the hint the way you suggested, but it doesn't seem obvious that the covers of the punctered disc extend uniquely. Besides, exactly as you mentioned, I would be more satisfied with the "étale homotopy type" statement. $\endgroup$ – Jędrzej Garnek Jun 25 '19 at 7:23
  • $\begingroup$ Minor comment, we have an equiv of etale ho-types from an n-disc to affine n-space. We'd like to "remove a point". The result is that in this case this is valid, it certainly isn't in general, it'd be nice to know when this sort of reasoning can be applied. $\endgroup$ – EBz Jun 25 '19 at 13:17
  • $\begingroup$ Have you tried asking the person who handed you the problem? $\endgroup$ – Steven Landsburg Jun 25 '19 at 14:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.