Some background

The (classical p-typical) de Rham-Witt complex is a complex of sheaves on a scheme over a perfect field of prime characteristic p.More precisely, it is a pro-system of differential graded algebras. In degree zero, it gives the Witt vectors and the first complex in the inverse limit is the de Rham complex. It provides an explicit way to compute crystalline cohomology. The constructions go back to Bloch,Deligne and Illusie. Since then various extensions and different methods are available. Current developements have applications in K-theory and p-adic Hodge theory.

Outline of the course

We will start with a self-contained introduction to Witt vectors. We continue with the definition of the (p-typical) de Rham-Witt complex following Illusie's paper and discusss the comparison to crystalline cohomology. Depending on the interest of the audience we will then decide in which direction to go: possible topics are the overconvergent de Rham-Witt complex of Davis-Langer-Zink, the big de Rham-Witt complex of Hesselholt-Madsen, logarithmic versions (Hyodo-Kato).
The lecture is scheduled twice per week. I plan to lecture on Mondays, and talk about examples, possible problems and exercises on Fridays. This might change during the semester.
I know it is a very early lecture, this is due to room scheduling problems. I appreciate anyone who comes anyway!

Lecture notes and problem sessions

I will update this page regularly with lecture notes, and exercises, and topics of coming lectures.
Note the time change of the lecture: now Monday/Friday 8-10 ct!

Continuing lecture notes.

Date	Title	Summary
12th October 2015	Introduction, Witt vectors (1)	Historical overview of the de Rham--Witt complex; definition of p-typical Witt vectors
16th October 2015	Witt vectors (2)	Frobenius, Verschiebung, restriction, relations etc.
19th October 2015	Witt vectors (3)	Big Witt vectors
23rd October 2015	Exercises (1)
26th October 2015	Crystalline Cohomology	A quick review
30th October 2015	Exercises (2)
2nd November 2015	de Rham-Witt complex (1)	The p-typical complex defined for Fp-algebras
6th November 2015	Exercises (3)
9th November 2015	de Rham-Witt complex (2)	Some properties.
13th November 2015	Exercises (4)
16th November 2015	de Rham-Witt complex (3)	Computation for group schemes Gm and Ga
20th November 2015	Exercises (5)
23th November 2015	de Rham-Witt complex (4)	Comparison with crystalline cohomology
27th November 2015	Exercises (6)
30th November 2015	Big de Rham-Witt complex (1)	Definition of big Witt complexes
4th December 2015	Exercises (7)
7th December 2015		Meeting of the GK in Windberg
11th December 2015	Big de Rham-Witt complex (2)	A few calculations.
14th December 2015	Big de Rham-Witt complex (3)	Two anticommutative graded algebras
18th December 2015	Exercises (8)
21st December 2015	Big de Rham-Witt complex (4)	Lambda-rings
8th January 2015	Exercises (9)
11th January 2015	No class this day.
15th January 2015	No exercises this day.
18th January 2015	Big de Rham-Witt complex (5)	Modules and derivations over lambda-rings (1).
22nd January 2015	Exercises (10)
25th January 2015	Big de Rham-Witt complex (6)	Modules and derivations over lambda-rings (2).
29th January 2015	Exercises 11
1st February 2015	Big de Rham-Witt complex (7)	Existence and uniqueness of the big de Rham-Witt complex.
5th February 2015	Exercises (12)

References

[Bou83] Nicholas Bourbaki: Algèbre commutative. Chapitre 8 et 9. (1983).

[CL98] Antoine Chambert-Loir: Cohomologie cristalline: un survol. http://www.math.u-psud.fr/~chambert/index.xhtml#research (1998)

[DLZ11] Christopher Davis, Andreas Langer and Thomas Zink: Overconvergent de Rham-Witt cohomology. Annales Scientifiques de l'Ecole Normale Supérieure,44(2), (2011).

[Hes05] Lars Hesselholt: Witt vectors. http://www.math.nagoya-u.ac.jp/~larsh/teaching/F2005_917/ (2005).

[HM03]Lars Hesselholt and Ib Madsen: The de Rham-Witt complex in mixed characteristic. http://www.math.nagoya-u.ac.jp/~larsh/papers/013 (2003).

[HK94] Osamu Hyodo and Kazuya Kato,: Semi-stable reduction and crystalline cohomology with logarithmic poles. Périodes p-adiques (Bures-sur-Yvette, 1988). Astérisque 223 (1994), 221-268.

[Ill79] Luc Illusie: Complexe de de Rham-Witt et cohomologie cristalline. Annales scientifiques de l'Ecole Normale Supérieure, 12(4): 501-661 (1979).

[LZ03] Andreas Langer and Thomas Zink: De Rham-Witt complex for a proper and smooth morphism. http://www.mathematik.uni-bielefeld.de/~zink/z_publ.html (2003)

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