why when one considers deformation functors for schemes in positive characteristic has one to define them over the ring of Witt vector?
Thanks
why when one considers deformation functors for schemes in positive characteristic has one to define them over the ring of Witt vector?
Thanks
If $X$ is a variety over $k$ a (perfect?) field of positive characteristic, then in general one doesn't have to define the deformation functor over $W$. It is actually an interesting question to compare $Def_X: Art_k\to Set$ and $Def_X: Art_W\to Set$. It is less common, but in theory you could take any $k$-algebra, $S$ and consider study the deformation functor $Def_X: Art_S\to Set$.
On the other hand, one often wants to know if a variety "lifts" to characteristic $0$. In this case, there are tons of $k$-algebras that you don't really care about because they aren't of characteristic $0$. You probably know that $W$ is a complete DVR of characteristic $0$ with residue field $k$, but also it is absolutely unramified.
Because of that last property one could think of this as a sort of "minimal" ring that one could consider deformations over and not miss any of the lifts to characteristic $0$. This is why in my mind it is a nice natural choice to study the deformation theory.