Let $X$ be a complex algebraic variety. Is the functor of the algebraic fundamental group $X\mapsto \pi_1^{alg} (X)$ the composition of the functor of the classical fundamental group $X\mapsto\pi_1(X_{ann})$ and the functor of profinite completion $G\mapsto \tilde{G}$?

Moreover, if we have the push-out diagram given by applying Van-Kampen Theorem to $U,V\subset X$ and we apply the profinite completion functor we get the push-out diagram of étale Van-Kampen Theorem?

Thanks in advance.

  • $\begingroup$ Some version of this statement should follow from Serre's GAGA principle (I think $X$ might need to be smooth and projective?). $\endgroup$ – Qiaochu Yuan Apr 3 '14 at 4:52
  • $\begingroup$ @QiaochuYuan Do you know whether the profinite completion functor is a left adjoint (see my answer below)? I recall this being true, but I honestly can't remember why. $\endgroup$ – Alex Youcis Apr 3 '14 at 5:18
  • $\begingroup$ @Alex: yes, I think so, although I'm a little afraid because I can't find an online reference confirming this. $\endgroup$ – Qiaochu Yuan Apr 4 '14 at 4:24

This was too long for a comment.

For the first question, we have the following:

Theorem: Let $X$ be a connected finite type scheme over $\mathbb{C}$. The functor which sends $Y\to X$ to $Y^\text{an}\to X^\text{an}$ induces an equivalence of categories between the category of finite etale covers of $X$, and the category of finite topological covers of $X^{\text{an}}$.

A proof of this theorem can be found in SGA 1, corollary 5.2

The hard part of this theorem is the Riemann existence theorem which, ultimately, amounts of the essential surjectivity of the functor.

Now, how does this answer your first question? For every $\mathbb{C}$-point $\overline{x}:\text{Spec}(\mathbb{C})\to X$ one obtains a point $x\in X^\text{an}$. The functor above then induces isomorphisms between the automorphism of group of the fiber functor $\text{Fib}_{\overline{x}}:\mathbf{Fet}/X\to\mathbf{Set}$ and the fiber functor $\text{Fib}_x:\mathbf{FinCov}/X\to\mathbf{Set}$ (where $\mathbf{Fet}/X$ is the category of finite etale covers, and $\mathbf{FinCov}/X$ is the category of finite topological covers). But, $\text{Aut}(\text{Fib}_{\overline{x}})=\pi_1^\text{et}(X,\overline{x})$ and $\text{Aut}(\text{Fib}_x)\cong \widetilde{\pi_1(X,x)}$.

As for your second question, I believe the answer should be yes because, if I recall, the profinite completion functor is a left-adjoint. This I am not sure of though.


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