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 $\mathrm{H}^2(X,\mathcal O_X)$ 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?

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?