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I have a very general, and possibly not very precisely stated question, which comes up quite often in my work, and I would be very happy to be able to address. To my dismay, I only have some very basic knowledge of algebraic geometry, so I cannot say whether this question is trivial or not, and I would appreciate any sort of advise anyone here might be able to give.

Question- Let $X$ be a scheme which is defined over a finite extension of $\mathbb F_p$. Is it, in general, possible to find a $\mathbb Z_p$-defined scheme such that $X\simeq Y\times_{Spec(\mathbb Z_p)}\mathbb F_p$?

Obviously, if $X$ is the reduction modulo $p$ of a given scheme over $\mathbb Z_p$, then this question has an affirmative answer. I was wondering, however, if this can be done in general, without any previous knowledge of the structure of $X$. Additionally, I would be interested to know if such a construction can be done in a factorial manner, and if it is known to conserve any properties of $X$? For example- if $X$ is an $\mathbb F_p$-group scheme, can $Y$ be constructed to be a group scheme over $\mathbb Z_p$?

Thank you.

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    $\begingroup$ You can define $X$ to be the composition $Y\to \text{Spec}\ \mathbb{F}_p$ and $\text{Spec}\ \mathbb{F}_p\to \text{Spec} \ \mathbb{Z}_p$. However, if you want $X$ to be flat over $\text{Spec}\ \mathbb{Z}_p$, this is often impossible. For instance, let $S_0$ be smooth and projective over $\text{Spec}\ \mathbb{F}_p$ of dimension $\geq 2$ such that $\Omega_{S_0/\mathbb{F}_p}$ is ample, and let $X$ be the blowing up of $S_0\times_{\text{Spec}\ \mathbb{F}_p} S_0$ along the graph of the Frobenius morphism. $\endgroup$
    – Jason Starr
    Commented Sep 19, 2017 at 11:42
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    $\begingroup$ There is an infinitesimal obstruction theory for infinitesimal deformations. The standard references are the two volumes by Illusie on the cotangent complex. $\endgroup$
    – Jason Starr
    Commented Sep 19, 2017 at 12:09
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    $\begingroup$ Related question: mathoverflow.net/questions/25337/… $\endgroup$
    – Daniel Loughran
    Commented Sep 19, 2017 at 17:18

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