I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.


$Hom(H^1\_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}\_1(X)/n$


Is there any reference for this? Why this is true?



Fundamental Group and Etale Cohomology
See Milne's online course notes on Étale Cohomology, Example 11.3, or Lei Fu's Étale Cohomology Theory, Proposition 5.7.20. (By passing to the direct limit over all $n$, you can even prove it for $\mathbf{Q}/\mathbf{Z}$.)


You only need $X$ to be connected Noetherian.


I am interested in alternative proofs not using torsors and Cech cohomology.


BTW, for $X$ normal, it is also true for $\mathbf{Z}$- or $\mathbf{Q}$-coefficients---both sides are $0$ in this case.


