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

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.

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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 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: 2008-Jan-10: Archive release 2007-Aug-11: Informal release.