Lemma 49.10.6. With notation as in Example 49.10.5 the schemes X_{n, d} and Y_{n, d} are regular and irreducible, the morphism Y_{n, d} \to X_{n, d} is locally quasi-finite and syntomic, and there is a dense open subscheme V \subset Y_{n, d} such that Y_{n, d} \to X_{n, d} restricts to an étale morphism V \to X_{n, d}.
Proof. The scheme X_{n, d} is the spectrum of the polynomial ring A. Hence X_{n, d} is regular and irreducible. Since we can write
f_ i = a_{i, (0, \ldots , 0)} + \sum \nolimits _{E \in T, E \not= (0, \ldots , 0)} a_{i, E} x^ E
we see that the ring B is isomorphic to the polynomial ring on x_1, \ldots , x_ n and the elements a_{i, E} with E \not= (0, \ldots , 0). Hence \mathop{\mathrm{Spec}}(B) is an irreducible and regular scheme and so is the open Y_{n, d}. The morphism Y_{n, d} \to X_{n, d} is locally quasi-finite and syntomic by Lemma 49.10.1. To find V it suffices to find a single point where Y_{n, d} \to X_{n, d} is étale (the locus of points where a morphism is étale is open by definition). Thus it suffices to find a point of X_{n, d} where the fibre of Y_{n, d} \to X_{n, d} is nonempty and étale, see Morphisms, Lemma 29.36.15. We choose the point corresponding to the ring map \chi : A \to \mathbf{Q} sending f_ i to 1 + x_ i^ d. Then
B \otimes _{A, \chi } \mathbf{Q} = \mathbf{Q}[x_1, \ldots , x_ n]/(x_1^ d - 1, \ldots , x_ n^ d - 1)
which is a nonzero étale algebra over \mathbf{Q}. \square
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