Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms

Question

Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?

Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a finite étale morphism. Let $A/S'$ be an abelian scheme. Then the following argument shows that the Weil restriction $\mathfrak{R}_{S'/S}$(A) (see section 7.6 of the book Néron Models by Bosch, Lütkebohmert and Raynaud) which a priori is just an fppf sheaf of abelian groups on $Sch/S$ is representable by an abelian scheme over $S$:

1) Theorem 1.5 of M. Olsson's paper Hom stacks and restriction of scalars (Duke Math. J. 134 (2006), 139-164.) gives a general criterion for a Weil restriction to be representable by an algebraic space, which applies here to show that $\mathfrak{R}_{S'/S}(A)$ is representable by an algebraic space over $S$.

By passing to a Galois cover, $\mathfrak{R}_{S'/S}(A)$ decomposes as a product of abelian schemes, so is representable by an algebraic space over $S$.

2) The arguments used to prove Proposition 7.6.5 (f) and (h) of Néron Models show that $\mathfrak{R}_{S'/S}(A)$ is smooth and proper as an algebraic space. Moreover the formation of the Weil restriction commutes with base change and its fibers are connected by Proposition A.5.9 of the book Pseudo-reductive Groups by Conrad,Gabber and Prasad.

3) By 1) and 2), $\mathfrak{R}_{S'/S}(A)$ is an abelian algebraic space in the sense of Section I.1 of the book Degeneration of abelian varieties by Chai and Faltings. By Theorem 1.9 of loc. cit. (due to Raynaud) it is actually an abelian scheme.

The "problem" with this proof is that 3) is a relatively delicate result. Of course, if $A$ is projective over $S$, the Weil restriction is automatically a scheme and we can do without 1) and 3). However, the question of when an abelian scheme is projective over the base is subtle in general: it is known to hold over a noetherian normal base but the argument is a key part of the proof of 3) anyway.

My motivation is to understand the push-forward functoriality of Deligne 1-motives over a base, and I would like to be able to consider general abelian schemes over general base schemes.

1) Does anyone know a simpler proof ?

2) If one assumes only that $S'/S$ is finite locally free, is $\mathfrak{R}_{S'/S}(A)$ (which might not be proper anymore) still a scheme ?

Answer 1

Let $X \to S'$ be a smooth projective morphism, and $S' \to S$ a finite étale morphism.

The Weil restriction $Y \to S$ of $X \to S'$ is a closed subscheme of the scheme of morphisms $S' \to X$ on $S$, which is projective; hence it is projective (this is Grothendieck's original construction).

Let us show that $Y \to S$ is again smooth. Construction of the Weil restriction is étale local on $S$; hence we may assume that $S' = S_1 \sqcup \dots \sqcup S_m$ is a disjoint union of $m$ copies $S_1, \dots, S_m$ of $S$. Call $X_i$ the restriction of $X$ to $S_i$. It follows readily from the definition that $Y \to S$ is the product $X_1 \times_S \dots \times_S X_m$, and this proves smoothness.

If $X \to S'$ is an abelian scheme, the group scheme structure of $X \to S'$ induces a group scheme structure on $Y \to S$, and this completes the proof.