4 problems

Let $k$ be the residue field of a discrete valuation ring $R$ of characteristic zero, and let $k[[z_2]]/k[[x]]$ be a cyclic $\Gamma$-extension with Galois sub-extension…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $\phi: Z \xrightarrow{} X$ be a cyclic $\Gamma$-cover of curves over $k$, and let $\phi_1: Y \xrightarrow{} X$…

Deformability in towers conjecture. If $G$ is cyclic, then the natural morphism

Strong liftability conjecture. The scheme $Y$ is a smooth projective scheme which is strongly liftable over $W_2(k)$.