## Witt vectors, lambda-rings, and arithmetic jet spaces

University of Copenhagen, April-June, 2016

This class will be an introduction to the theory of Witt vectors, lambda-rings, and arithmetic jet spaces, which are in the end different points of view on the same mathematics. We will spend most of the time on the usual 'big' and 'p-typical' theory, but I hope to spend some time on a new 'elliptic' analogue of the usual theory.

In the first lecture, I'll give a transparent and conceptual definition of the p-typical Witt vector functor which was first found by Joyal and is not very well known. The lectures after that will be on the following basic topics: the connection between Witt vectors and lambda-rings, variants (big, ramified, relative to number fields, etc), and algebraic-geometric properties which allow one to extend Witt vectors and lambda-structures from rings to schemes. I'll also try to take an approach which makes the development as inevitable as possible. So a lot of attention will be paid to naturality and choices.

To follow the class, you should be comfortable with category theory and basic commutative algebra. It would be helpful to have some familiarity with scheme theory and algebraic number theory, but it's not essential.

Lectures notes

Here. Taken by Lars Hesselholt, except for one lecture by Amalie Høgenhaven

Lectures, with supplementary references:

1. p-derivations and Witt vectors
2. Arithmetic differential operators and the Witt components
3. Proof of Joyal's theorem, finite length Witt vectors, and the general theory of unary operations on rings
4. Examples of birings, the category of P-rings, reconstruction and recognition
5. Examples of composition algebras, the enveloping principle, table of analogies between module theory and commutative algebra
6. Inverting Frobenius, the Necklace theorem
7. Proof of the Necklace theorem, the necklace coordinates on Witt vectors
8. Consequences of the Necklace theorem, Verschibung, and Teichmueller representatives
9. Frobenius lifts in general, construction of W
10. Frobenius lifts in general, construction of W (continued)
11. Frobenius lifts in general, representability of W
12. Frobenius lifts in general, representability of W (continued). Classical structures in general contexts.
13. Geometry, the F1 picture, big Witt vectors and lambda-rings (an un-prepared lecture)
14. Witt vectors and semirings (another un-prepared lecture)
Problems for exercise classes: