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Let X be a variety over a finite field.

Let $x$ be a closed point of $X$. Then from it we are supposed to get an automorphism $Frob_x$ that induces a map from the group of zero cycles to the abelianization of the etale fundamental group. What is this $Frob_x$? I don't think it is the usual Frobenius, since it depends on a closed point some how and is supposed to produce a etale cover (really, a coset of etale covers).

For context, I was reading this survey here:https://www.mathi.uni-heidelberg.de/~schmidt/papers/schmidt-luminy-2013-revised.pdf

page 4.

It didn't seem to be any of the version of Frobenius on wikipedia.

I guess it could be related to the residue field of x...

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The Frobenius automorphism $\mathrm{Frob}_x$ doesn't induce the morphism $$C^0(X) \to \pi_1(X)^{ab}.$$ Rather, this morphism is defined by sending a closed point $x$ to the associated Frobenius element $\mathrm{Frob}_x \in \pi_1(X)^{ab}$.

This is a version of the Artin map from class field theory.

More precisely, if $Y \to X$ is etale and Galois, and if $y\in Y$ lies over $x$, then there is a canonical element $\mathrm{Frob}_y \in \mathrm{Gal}(Y/X),$ characterized by the fact that it fixes the point $y$, and induces the Frobenius automorphism on the extension of residue fields $\kappa(y)/\kappa(x)$. If we choose a different point $y'$ lying over $y$, then $\mathrm{Frob}_{y'}$ is conjugate to $\mathrm{Frob}_y$ in $\mathrm{Gal}(Y/X)$, and hence the image of $\mathrm{Frob}_y$ in $\mathrm{Gal}(Y/X)^{ab}$ depends only on $x$, and is denoted $\mathrm{Frob}_x$.

If we pass to the inverse limit over all $Y$, we obtain a well-defined element $\mathrm{Frob}_x \in \pi_1(X)^{ab}$.

The review you're reading is about geometric class field theory, and Theorem 7, which is stated soon after the discussion you asked about, is a generalization to higher dimensional varieties of the usual Artin reciprocity law for function fields of curves over finite fields (the function field case of classical class field theory).