I thought of this several months ago and forgot about it. Now I rethought of it again and I just can't find it anywhere in the literature, so I'll ask here.


Is it known whether or not there exists a (smooth, proper, ...) variety over a field $k$ (perfect? alg. closed?) of positive characteristic that lifts to characteristic $0$ over some ramified extension of $W(k)$ and also lifts to $W\_2(k)$, but does not lift over $W(k)$ itself?


In other words, if it lifts to characteristic $0$ (in a way related to $W(k)$) and it lifts to $W\_2(k)$ must it lift via $W(k)$ itself?


I have looked at examples that lift to $W\_2(k)$ but not to $W\_3(k)$ and hence not over $W(k)$, but they don't seem to lift to char $0$ at all. I've also looked examples that lift over a ramified extension but not $W$, but these can easily be shown to not lift to $W\_2$ as well.



Lifting to Characteristic 0 not over W
Theorem 1.1, M1a of the following article of Ravi Vakil,
"Murphy's Law in algebraic geometry: Badly-behaved deformation spaces", Invent. Math. 164 (2006), 569--590, available at [<http://math.stanford.edu/~vakil/files/Mjul0705.pdf>](http://math.stanford.edu/~vakil/files/Mjul0705.pdf).


Apply this theorem to the "germ" of $\text{Spec}(\mathbb{Z}[x]/\langle p(x^2-p) \rangle)$ over $\text{Spec}(\mathbb{Z})$. 


