The de Rham-Witt complex

Content/Literature/Recommended previous knowledge: 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. 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). Literature: [Bou83] Nicholas Bourbaki: Alèbre commutative. Chapitre 8 et 9. (1983). [CL98] Antoine Chambert-Loir: Cohomologie cristalline: un survol. http://perso.univ-rennes1.fr/antoine.chambert-loir/publications.html (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.htm (2003) Prerequisits: algebra.