Let $X\rightarrow Y$ a ramified double cover of smooth projective curves, and let $$\mathcal G:=Res_{X/Y}(SL_n)$$ be the Weil restriction of the constant group scheme $SL_n$ over $X$.

Question: Is $\mathcal G$ flat over $Y$ ?

N.B: I know (c.f. Néron Models) that $\mathcal G$ is flat over $Y$ if the cover is étale.

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    $\begingroup$ Weil restriction through a finite flat map between noetherian schemes always preserves smoothness, so $\mathscr{G}$ is even $Y$-smooth. This is an application of the infinitesimal criterion, as is shown in the proof given in the reference you mention (where the expression "without any further assumptions" in their formulation of Prop. 5 of section 7.6 does not entail the etaleness from the preceding line in the text, as you can see when reading the proof). $\endgroup$ – nfdc23 Mar 14 '16 at 18:03

As in the commentry of of @nfdc23, $\mathcal G$ is even smooth. Ref: Néron Models, Prop. 5 of section 7.6.


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