### The basic geometry of Witt vectors, I: The affine case

**Abstract:**

We give a concrete description of the category of étale
algebras over rings of Witt vectors of finite length. We allow not just the usual,
*p*-typical Witt vectors, where *p* is a prime
number, but also those taken with respect to arbitrary sets of primes in rings of integers
in global fields. This includes rings of “big” Witt vectors.
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 traditional big and *p*-typical Witt vectors.
This paper was written 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.

Download: pdf

The paper's page at the arxiv.org.

The paper's page
at Journal of Algebra and Number Theory.

#### Release notes

2010-Oct-14: Accepted, Algebra & Number Theory.

2010-Oct-07: Probably the final version. There is a more detailed introduction. There
have also been a few minor corrections and improvements to the body of the paper.

2010-Jun-02: The version at arxiv.org has been updated.

2010-May-14: The paper has been cut in half for publication. The first half is above, and
the second half is
here.
The introduction to this half has been completely rewritten.
Otherwise, the only changes are some minor editing.

2009-Jun-17: Major changes to the first half. The proof of the main theorem has been changed.
The exposition now has much more detail and its organization is much tighter.

2008-Aug-03: I added some detail on the relation to Buium's functor and cut
out everything about set theory. Many other minor changes.

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-11: Informal release.