  We give a concrete description of the category of etale algebras over the
ring of Witt vectors of a given finite length with entries in an arbitrary
ring. We do this not only for the classical p-typical and big Witt vector
functors but also for variants of these functors which are in a certain sense
their analogues over arbitrary local and global fields. The basic theory of
these generalized Witt vectors is developed from the point of view of commuting
Frobenius lifts and their universal properties, which is a new approach even
for the classical Witt vectors. The larger purpose of this paper is to provide
the affine foundations for the algebraic geometry of generalized Witt schemes
and arithmetic jet spaces. So the basics here are developed somewhat fully,
with an eye toward future applications.
