In the paper, Notes on etale cohomology of number fields, Mazur insists that for a finite **etale** morphism $\pi$, the Norm theorem holds (on. p.543, Remarks (e)). 


More specifically, let $\pi : Y \rightarrow X$ be a finite morphism, where $Y=\text{Spec}(D\_Y)$ and $X=\text{Spec} (D\_X)$, and $D$'s are Dedekind domains. For a constructible sheaf $F$ on $Y$, there are the Norm maps
$$
N\_r : \text{Ext}^r\_Y (F, \mathbb{G}\_m) \rightarrow \text{Ext}^r\_X (\pi\_\* F, \mathbb{G}\_m)
$$
(on. p.542 of the same paper). If $\pi$ is etale, then $N\_r$ is an isomorphism for any $r$. Mazur commented that a beautiful morphism $F \rightarrow \pi^\*\pi\_\*F$ provides an inverse to $N\_r$. 


Can someone explain this in details?



Norm theorem for finite etale morphisms between Dedekind affine schemes
One has an isomorphism 
$$N\_r: {\rm{Ext}}^r\_Y(F, \pi^{\ast}G) \rightarrow {\rm{Ext}}^r\_X(\pi\_{\ast}F, G)$$
for finite etale $\pi:Y \rightarrow X$ between arbitrary schemes and any abelian etale sheaves $F$ on $Y$ and $G$ on $X$, without needing to appeal to an explicit procedure for an inverse map (though the argument below adapts to show that it is given by composing $\pi$-pullback with a canonical map $F \rightarrow \pi^{\ast}\pi\_{\ast}F$ to which Mazur refers). 


The idea is to reduce the problem to a sheaf-theoretic isomorphism, so one can then work locally on the base to pass to the case of a split covering that is easy. (This uses that we got away from the original setting in which $Y$ is required to be irreducible.) 


Let's first recall how $N\_r$ is defined (in the spirit of coherent duality). Since $\mathbf{R}{\rm{Hom}}\_Y = \mathbf{R}\Gamma\_Y \circ \mathbf{R}\mathscr{H}om\_Y$ and $\Gamma\_Y = \Gamma\_X \circ \pi\_\*$ with $\pi\_\*$ exact (as $\pi$ is finite), we define $N\_r$ via passage to $r$th homology on the effect of applying
$\mathbf{R}\Gamma\_X$ to the natural sheaf map
$$\pi\_{\ast}\mathbf{R}\mathscr{H}om\_Y(F, \pi^{\ast}(\cdot)) \rightarrow
\mathbf{R}\mathscr{H}om\_X(\pi\_{\ast}F, \pi\_{\ast}\pi^{\ast}(\cdot))
\rightarrow \mathbf{R}\mathscr{H}om\_X(\pi\_{\ast}F, \cdot)$$
using the sheaf-theoretic trace for finite etale maps (or even just quasi-finite etale separated maps) at the final step.


Thus, to show $N\_r$ is an isomorphism for finite etale $\pi$ it suffices to show that the above displayed map is an isomorphism. But this task is of etale-local nature on $X$, and since $\pi$ is finite etale we can pass to an etale cover on $X$ to arrive at the case that $\pi$ is a split covering; i.e., $Y$ is a finite disjoint union of copies $X\_1, \dots, X\_n$ of $X$. Now $F$ amounts to a collection of sheaves $F\_1, \dots, F\_n$ on the $X\_i$'s and the map in question is the natural composite map
$$\oplus\_i \mathbf{R}\mathscr{H}om\_X(F\_i, G) \rightarrow
\mathbf{R}\mathscr{H}om\_X(\oplus\_i F\_i, \oplus\_i G)
\rightarrow \mathbf{R}\mathscr{H}om\_X(\oplus\_i F\_i, G)$$
(using the addition map $\oplus\_i G \rightarrow G$ at the final step) for any abelian etale sheaves $F\_1, \dots, F\_n, G$ on $X$. But this is obviously an isomorphism.


