The basic geometry of Witt vectors
Abstract:
This is a foundational account of the étale topology of
generalized Witt vectors.
The theory of the usual, “p-typical” Witt vectors of p-adic schemes
of finite type is already reasonably well developed.
The main point here is to generalize this theory in two different ways.
We allow not just p-typical Witt vectors but also, for example, those taken
with respect to any set of primes in any
ring of integers in any global field. In particular, this includes
the “big” Witt vectors.
We also allow not just p-adic schemes of finite type
but arbitrary algebraic spaces
over the ring of integers in the global field.
We give similar generalizations of Buium's formal arithmetic jet spaces.
We establish concrete geometric descriptions of Witt spaces and arithmetic
jet spaces and investigate whether many standard geometric
properties are preserved by these functors.
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Release notes
2008-Mar-22: Very minor changes.
2008-Mar-16:
- Section 9 has been added. It gives a geometric construction of arithmetic jet spaces.
- The term "Greenberg transform" has been changed to "arithmetic jet space". Buium pointed
out that his jet space functor is the p-adic completion of mine and
that the functor defined by Greenberg is the special fiber.
- Many other minor improvements.
- The introduction has been slightly rewritten to take these things into account.
2008-Jan-10: Archive release
- All the Lambda-algebraic geometry has been removed and will appear in a future paper,
Sheaves in Lambda-algebraic geometry.
- Section 1 now begins with a mini-essay on the defining Witt vectors.
2007-Aug-??: Informal release.