# Existence of flat models of a smooth finite type algebra over $R((t))$

Let $k$ be a field, $R$ a $k$-algebra (of finite type if necessary), $B$ an algebra of finite type over ring of the formal Laurent 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 allow this?

• 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
• 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