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Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary!), $B$ an algebra of finite type over ring of formal Laurant series $R((t))$, which is smooth.

up to this generality, can one construct a flat model of $B$ over ring of formal power series $R[[t]]$ (i.e. a flat algebra $\tilde{B}$ over $R[[t]]$ such that $\tilde{B}\otimes_{R[[t]]}R((t))=B$)? if not what could be the weakest assumption that will serve it?

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Why not $\tilde{B}=B$? –  Laurent Moret-Bailly Dec 14 '10 at 12:05
    
by model I mean that is really defined over entire Spec R[[t]], to be more precise I would require "faithfully flatness"... –  Samuel Dec 14 '10 at 13:40
1  
Then you should at least assume $B$ faithfully flat over $R((t))$. –  Laurent Moret-Bailly Dec 14 '10 at 14:01
    
of course, you are right! –  Samuel Dec 14 '10 at 14:43

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