# What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?

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Two things: (i) I'm guessing you want examples of varieties which do not lift, since it is easy to give examples which do. (ii) If you don't put any further conditions on the problem, it seems to me to be trivial: if you have a finite set of equations with coefficients in $\mathbb{F}_q$, you can lift them, coefficient by coefficient, to a set of equations with coefficients in the ring of integers of some number field. Doesn't this show what you want? –  Pete L. Clark May 5 '11 at 3:45
Thanks Pete, I will edit the question. As for (ii), I certainly want to avoid things like Z_p[x]/(px^2 + x) lifting X = Spec F_p (actually, I care most about the case of lifting a finite scheme over F_p to a finite scheme over Z_p). –  David Zureick-Brown May 5 '11 at 3:54
@David: okay, now you have edited to a question that I don't know how to answer. :) –  Pete L. Clark May 5 '11 at 4:00

I received the following very explicit answer via private communication: the algebra

$$A = \mathbb{F}_p[x_1,\ldots,x_6]/(x_1^p,\cdots, x_6^p, x_1x_2 + x_3x_4 + x_5x_6)$$

does not lift to a finite flat $\mathbb{Z}_p$ algebra. (I am still working out the details of why this does not lift.) This is exampe 3.2(4) of Berthelot-Ogus.

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