# The étale fundamental group as a functor

The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continuous map $f\colon (X,x)\rightarrow (Y,y)$ induces a homomorphism $f_* \colon \pi_1 (X,x)\rightarrow \pi_1 (Y,y)$ via composition with $f$ i.e. $[\alpha]\mapsto [f\circ\alpha]$ where $\alpha$ is a loop in $X$ based at $x$.

I know that the étale fundamental group of a variety over $K$ with a distinguished $K$-point is supposed to be functorial in the same way (i.e. gives a functor from the category of pointed varieties over $K$ to the category of groups) but there are two technical issues I need to settle. Here is my definition of the étale fundamental group:

Let $X$ be a variety over $K$, let $x\in X(K)$ and let $\textbf{Cov}(X,x)$ denote the category of finite pointed étale covers $(Y,y)\rightarrow (X,x)$ where $y\in Y(K)$. Let $\Phi_x\colon \textbf{Cov}(X, x)\rightarrow \textbf{Set}$ denote the fibre functor sending a covering $\phi: (Y,y)\rightarrow (X,x)$ to the fibre $\phi^{-1}(x)\subseteq Y$. Then the étale fundamental group of $(X,x)$ is defined as $\pi_1^{\text{ét}} (X,x) = \operatorname{Aut}(\Phi_x)$.

Now I want to show that if $f\colon (X,x)\rightarrow (Y,y)$ is a pointed morphism of varieties, it induces a homomorphism $f_* \colon \pi_1^{\text{ét}} (X,x)\rightarrow \pi_1^{\text{ét}} (Y,y)$. I read right at the start of these notes how to begin: pick a cover $\phi\colon (Y', y')\rightarrow (Y,y)$ in $\textbf{Cov}(Y,y)$. Then apparently the pullback $\phi^*\colon X\times_Y Y'\rightarrow X$ is a finite étale cover of $X$ (i.e. an object in $\textbf{Cov}(X,x)$) and the fibre of $\phi^*$ above $x$ is isomorphic to the fibre of $\phi$ above $y$.

My questions are:

1. Why is $\phi^* \colon X\times_Y Y' \rightarrow X$ a finite étale cover of $X$?
2. How can I show the claim that $\Phi_x (X\times_Y Y', \phi^*) \cong \Phi_y (Y', \phi)$?

I will mainly be restricting my varieties $X$ and $Y$ to be elliptic curves, so an explanation assuming this would also be very welcome.

Edit: Kevin Carlson's comments below have helped me solve question 2 by myself. I include a proof here for completeness, but because I still don't have an answer to question 1 I have not posted this as an answer.

To get a canonical isomorphism $\Phi_x (X\times_Y Y')\cong \Phi_y (Y)$, note that $\Phi_x (X\times_Y Y') = \text{Spec}(K)\times_X (X\times_Y Y')$ and $\Phi_y (Y') = \text{Spec}(K)\times_Y Y'$ where $X\xleftarrow{x}\text{Spec}(K)\xrightarrow{y} Y$ are the $K$-points. Then since $f\circ x = y$, we have a commutative diagram where the two inner squares are pullbacks:

Then from the pullback pasting lemma (see the section "pasting of pullbacks") the outer rectangle is a pullback diagram and so it follows that $\Phi_x (X\times_Y Y')$ has the UMP of the fibre product $\text{Spec}(K)\times_Y Y' = \Phi_y (Y)$, so must be isomorphic.

• Have you made any effort to check these things? How comfortable are you with fiber products of schemes? The second question, for instance, is exactly what fiber products are for. Jan 25, 2016 at 15:06
• @KevinCarlson I'm not particularly confident with fibre products of schemes past their definition, although I have made an attempt using their universal properties to answer these two things. I will try again on the second question if it follows from general facts about fibre products in any category. The first question, with the étale assumption, seems more tricky but I might be missing something obvious. Jan 25, 2016 at 16:01
• @KevinCarlson for my second question specifically, I have been able to get a unique morphism $\Phi_x (X\times_Y Y') \rightarrow \Phi_y (Y')$ but don't see how to get one in the other direction. Jan 25, 2016 at 16:13
• I'm not claiming it's all obvious, just seeing where you're at. The thing about the fibers does follow from the pullback pasting lemma when you see the scheme-theoretic fiber as the pullback over a map from the spectrum of $K$. Jan 25, 2016 at 16:42
• You should be able to find in standard references that the class of finite (resp. étale, surjective) morphisms is closed under pullback. For instance, try the Stacks project. Jan 25, 2016 at 19:19