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2d

comment 
How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
You're perfectly right that the assumptions of condition (ii) of sga3 exp.viB Cor.4.4 are met. The point I was worried about is whether (ii) should imply (i) in loc. cit., and it is quite striking that for reduced $S$ (as in your case) this implication holds. Apologies! 
2d

comment 
How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
Maybe could you have a look at Lemma 4.2.8 in the paper S. Brochard, Foncteur de Picard d’un champ algébrique, Math. Ann. 2009. Also note that this has been extended to alg. stacks in an Erratum to the paper, available on the author's webpage. 
2d

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How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
Dear Heer, do you in fact assume that $G$ is smooth over $S$? If I'm not mistaken, with your assumptions it could be that $G=G_s\amalg G_t$ mapping to $S$ in the natural way, which I doubt you want to consider. 
Feb 25 
comment 
Implicit Function Theorem on Singular Varieties
Dear Giulio, in Algebraic Geometry the property of finiteness has a specific meaning which implies properness. The example of Laurent is bijective but not proper since the image of the closed subset $X_2$ is not closed in $Y$. 
Feb 18 
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$R^{\dim X\dim Y}f_{\ast}\omega_X \simeq \omega_Y$ in positive characteristic?
Isn't such vanishing the result of Chatzistamatiou and Rülling in "Higher direct images of the structure sheaf in positive characteristic", ANT 2011? 
Feb 18 
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Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?
Why KeelMori? The original Artin theorem proves that the quotient fppf sheaf, which since there are no stabilizers is also the quotient stack [X/E], is an algebraic space. See e.g. Artin, The implicit function theorem in algebraic geometry, Thm 7.1. 
Dec 22 
awarded  Popular Question 
Dec 19 
comment 
How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?
In your first sentence, the ptorsion kernel $E[p]$ of a supersingular elliptic curve is not $\alpha_p$. Indeed, the former has order $p^2$. 
Dec 9 
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Quotient of a (nonlinear) algebraic group by a closed subgroup
Also Perrin, Approximation des schémas en groupes quasicompacts sur un corps, Bull. SMF 1976, shows the following: if G is quasicompact and H is either a normal subgroup, or defined by a finitely generated ideal, then the fpqc sheaf G/H is representable by a scheme. 
Dec 3 
answered  Regular rings and formally smooth algebras 
Nov 16 
revised 
Is this formally étale morphism of schemes an isomorphism?
deleted 193 characters in body 
Nov 15 
revised 
Is this formally étale morphism of schemes an isomorphism?
added 5 characters in body 
Nov 15 
answered  Is this formally étale morphism of schemes an isomorphism? 
Nov 8 
comment 
Is this formally étale morphism of schemes an isomorphism?
Two questions: 1) I'd be happy to have details on the proof that $\mathfrak{m}'=\mathfrak{m}B$. 2) For formal glueing in the very last step, the quoted lemma from the Stacks Project requires to know that $B$ is a finite $A$module, how do you deal with this issue? 
Nov 4 
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Fantastic properties of Z/2Z
That is terrific! Do you know a reference where I can read more on this? 
Nov 3 
accepted  Is this formally étale morphism of schemes an isomorphism? 
Nov 3 
comment 
Is this formally étale morphism of schemes an isomorphism?
OK, I get it. Thank you for this nice contribution. The particular situation that I have in mind has additional features (like quasicompactness) that your example doesn't, but it helped me anyway to understand things better. Thanks again! (And if you happen to have ideas in the quasicompact case...) 
Nov 3 
revised 
Is this formally étale morphism of schemes an isomorphism?
Added assumptions to the original question to fit with the actual situation I'm interested in. 
Nov 3 
comment 
Is this formally étale morphism of schemes an isomorphism?
Why it is true that $f$ is an iso on local rings at points of $Y$? 
Nov 3 
comment 
Is this formally étale morphism of schemes an isomorphism?
... and $S$ is noetherian. 