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Let $X$ be a scheme with base point $x:\operatorname{Spec}(K)\rightarrow X$ and algebraic fundamental group $\pi_1(X, x)$. Let $H$ be a normal subgroup of $\pi_1(X,x)$. How do we construct a scheme $X_H$ and a morphism $p:X_H\rightarrow X$ such that $p_{*}(\pi_1(X_H))=H$ where $p_{*}$ is the induced map on fundamental groups?

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By algebraic fundamental group do you mean the profinite etale fundamental group? Are you assuming that $H$ is normal and open? If so, then $X_H$ exists essentially by the definition of $\pi_1(X,x)$. – Matt E Sep 8 '11 at 14:21
Thanks for the response. By algebraic fundamental group I mean the automorphism group of the fibre functor on the etale site of $X$ at the base point $x$. I'd like to know how to construct $X_H$ given $X$ and $H$. – Krasnicki Sep 13 '11 at 11:40
Dear Krasnicki, Do you really mean the etale site (which include all etale maps, finite or not), or do you mean the category of finite etale covers, as in SGA 1. (The latter is what is usually involved when people speak of the etale, or algebraic, fundamental group, and it is a then a profinite group, just as Galois groups are profinite groups.) If you are not sure which you mean, perhaps you could give some indication of the context in which this question is arising. Regards, – Matt E Sep 13 '11 at 11:59
The $\pi_1$ that I have in mind is the one in SGA1. Sorry for the confusion. – Krasnicki Sep 13 '11 at 12:14

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