3
$\begingroup$

why when one considers deformation functors for schemes in positive characteristic has one to define them over the ring of Witt vector?

Thanks

1 Answers 1

1

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.