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 ?

doget an abelian scheme. Hopefully descent now applies to conclude that the restriction of scalars itself was an ab. scheme. Regards, Matthew $\endgroup$ – Emerton Apr 10 '13 at 18:07