Let $X$ be a smooth proper geometrically integral scheme over $\overline{\mathbb F\_p}$. Assume $X$ is the specialization of a smooth proper scheme over $\mathbb Z\_p^{nr}$. Let $L$ be an ample line bundle on $X$.


I would like to show that $L^{\otimes p}$ lifts to characteristic zero. However, the obstruction to lifting $L$ lives in $\mathrm{H}^2(X,\mathcal O\_X)$ which might be non-zero. 


On the other hand, the obstruction space is an $\overline{\mathbb{F}\_p}$-vector space. So the obstruction vanishes after multiplying with $p$.


Does this imply that $L^{\otimes p}$ lifts?



Lifting line bundles
This cannot always be done. If $X$ is a supersingular K3 surface then there are $22$ ample line bundles $L$
that give independent classes; if every $L^{\otimes p}$ lifted to char. zero, then you'd have a K3 surface in char. zero with $22$ independent line bundles, which is impossible. (Or $X$ could be any liftable unirational surface with $p\_g(X)>0$.)


