An attempt to compactify the unified Kummer-Artin-Schreier-Witt theory

Abstract. We could constract the group schemes ${\cal W}_n$ which give the deforamtions of the group scheme of Witt vectors of length $n$ to the torus ${\Bbb G}_m^n$. By using them, we could constract the unified Kummer-Artin-Schreier-Witt Theory. If we want to describe the ramified cyclic coverings in mixed characteristic case bu using this theory, we need to compactify the group schmes in a suitable way. We will give a candidate for the compactification in $n = 2$ case by using ruled surfaces.