Talk:Étale fundamental group

Untitled
I would suggest to remove the link to Topological space from the "See also" section. In my opinion it has nothing to do with Étale fundamental group more than with any other topic in topology or geometry. —Preceding unsigned comment added by 134.58.253.57 (talk) 12:46, 12 January 2011 (UTC)


 * Agreed. And from my understanding this topic actually has less to do with topological spaces than most topics in topology and geometry.  Removing the link. Tony Beta Lambda (talk) 09:15, 24 March 2018 (UTC)

Definition of F as Yoneda functor
Is F $$X^{Opp} \rarr Set^X$$, depending on $$x$$, or $$Schemes^{Opp} \rarr Set^{Schemes}$$ depending on $$X$$? ᛭ LokiClock (talk) 18:32, 26 December 2012 (UTC)

Definition of Galois covers and clarification needed
The article has written:


 * The functor $$F$$ is not representable, however, it is pro-representable, in fact by Galois covers of $$X$$. This means that we have a projective system $$\{X_j \to X_i \mid i < j \in I\}$$ in $$C$$, indexed by a directed set $$I,$$ where the $$X_i$$ are Galois covers of $$X$$, i.e., finite étale schemes over $$X,$$ such that $$\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X)$$.

But this means


 * The functor $$F$$ is not representable, however, it is pro-representable, in fact by Galois covers of $$X$$. This means that we have a projective system ... where the $$X_i$$ are Galois covers of $$X$$ such that $$\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X)$$.

Keep in mind that Galois cover redirects here, so it's considered undefined outside of this article.

Should it instead say this?


 * i.e., finite étale schemes over $$X$$ such that $$\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X)$$

Because that would mean the article is actually saying


 * The functor $$F$$ is not representable, however, it is pro-representable, in fact by Galois covers of $$X$$. A Galois cover is a finite étale scheme over $$X$$ such that $$\#\operatorname{Aut}_X(X_i) = \operatorname{deg}(X_i/X)$$.

If this was not the intention, and the agreement of the degree and number of automorphisms is not true solely because we have a projective system of Galois covers, but due to the details about what pro-representability means (no definition is linked to), then since what follows "this means" can be interpreted as giving the definition of pro-representability by Galois covers to begin with, not listing arbitrary implications of this particular Yoneda functor $$F$$ being pro-representable, I would say it is potentially misleading to state the agreement between degree and number of automorphisms in this way, rather than in its own sentence with its derivation made explicit. At the same time it would be better to say outright that pro-representability by Galois covers is being defined, rather than let readers mistake that definition for a statement of arbitrary properties of $$F$$ and become confused about what statements the article is really making. ᛭ LokiClock (talk) 06:09, 14 September 2015 (UTC)


 * I agree with you. Go ahead and improve it, ideally with precise references, so that interested readers can check with the literature themselves! I am busy right now, but Milne's "Etale cohomology" §I.5 has a concise statement of the results, but no proofs. Jakob.scholbach (talk) 09:09, 14 September 2015 (UTC)


 * I don't have access to that, but "Lectures on Étale Cohomology" covers this as well. To rework the language I would want to examine it carefully, but I don't have pen and paper right now. ᛭ LokiClock (talk) 01:35, 15 September 2015 (UTC)

Assessment comment
Substituted at 02:42, 5 May 2016 (UTC)

The notation $$\operatorname{Hom}_X(x, Y)$$ is never introduced
The notation $$\operatorname{Hom}_X(x, Y)$$ is never introduced. No connection is made to the category $$C$$. Olivierbbb (talk) 11:48, 15 October 2023 (UTC)