Example 66.18.2. Let $K$ be a characteristic $0$ field endowed with an automorphism $\sigma$ of infinite order. Set $Y = \mathop{\mathrm{Spec}}(K)/\mathbf{Z}$ and $X = \mathbf{A}^1_ K/\mathbf{Z}$ where $\mathbf{Z}$ acts on $K$ via $\sigma$ and on $\mathbf{A}^1_ K = \mathop{\mathrm{Spec}}(K[t])$ via $t \mapsto t + 1$. Let $Z = \mathop{\mathrm{Spec}}(K)$. Then $W = \mathbf{A}^1_ K$. Picture

$\xymatrix{ \mathbf{A}^1_ K \ar[r]_ q \ar[d]_ p & \mathop{\mathrm{Spec}}(K) \ar[d]^ g \\ \mathbf{A}^1_ K/\mathbf{Z} \ar[r]^ f & \mathop{\mathrm{Spec}}(K)/\mathbf{Z} }$

Take $x$ corresponding to $t = 0$ and $z$ the unique point of $\mathop{\mathrm{Spec}}(K)$. Then we see that $F_{x, z} = \mathbf{Z}$ as a set.

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