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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asked May 5, 2011 at 2:59
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$\begingroup$ 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? $\endgroup$– Pete L. ClarkCommented May 5, 2011 at 3:45
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$\begingroup$ 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). $\endgroup$– David Zureick-BrownCommented May 5, 2011 at 3:54
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4$\begingroup$ @David: okay, now you have edited to a question that I don't know how to answer. :) $\endgroup$– Pete L. ClarkCommented May 5, 2011 at 4:00
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2 Answers
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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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$\begingroup$ have you worked them out, the details? $\endgroup$– user140761Commented May 20, 2019 at 5:23
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Am I missing something or is this the classical question of Serre? A class of examples is given in Exemples de variétés projectives en caractéristique p non relevables en caractéristique zéro. Proc. Nat. Acad. Sci. U.S.A. 47 1961 108–109.
They come from the quotient of some complete intersections by some finite groups, but if you read closely the proof (by the theory of the étale fundamental group, the impossibility of constructing a lift is reduced to the impossibility of constructing some group representations), you see that the ideas are in fact quite general. In order to give a non vacuous answer, let me also draw your attention to the letter of Serre in the appendix of the document illusie_trieste.pdf on Luc Illusie's website.
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3$\begingroup$ I was under the impression that Serre's constructions have dimension at least 2, and David Brown is looking for something finite. $\endgroup$– S. Carnahan ♦Commented May 5, 2011 at 19:24