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Let $X$ be a affine smooth scheme finite type over $A/pA$, where $A$ a complete DVR and $chk=p>0$.

I know that since $H^{2}=0=H^{1}$, we have a unique lifting to $A/p^{2}$. In algebraic schemes case, we can obtain a trivial deformation by fiber product. But in this case, for example, $A$ is a p-adic number ring, What is the trivial deformation over $A/p^{2}$ ?

PS: I added some smooth conditions to $X$.

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  • $\begingroup$ There might not exist a lifting of $X$ to $A/p^2$, let alone a canonical one. $\endgroup$
    – Angelo
    Commented Apr 6, 2013 at 9:05
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    $\begingroup$ I'm confused Angelo. The question says $X$ is affine and the obstruction to such a lifting lies in $H^2$ which vanishes, so doesn't such a lift always exist? $\endgroup$
    – Matt
    Commented Apr 6, 2013 at 16:15
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    $\begingroup$ @Matt: Assuming the scheme is finite type over $k$, the obstruction most naturally lives in an Ext group of the cotangent complex. If $X$ is smooth, this reduces to $H^2$, which will be zero as you say. However, this need not be the case if $X$ is singular. $\endgroup$
    – Jason Starr
    Commented Apr 6, 2013 at 16:22
  • $\begingroup$ Ah, thanks! I've clearly been indoctrinated by only thinking about liftability of smooth varieties. $\endgroup$
    – Matt
    Commented Apr 6, 2013 at 19:40

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