The de Rham-Witt complex was introduced by Bloch and Deligne-Illusie in the late 1970's. Their main results were firstly that the Zariski cohomology of this complex computes Grothendieck-Berthelot's crystalline cohomology of a smooth scheme over a perfect field of positive characteristic, and secondly that if X is additionally proper, then the slope decomposition à la Dieudonné-Manin of the underlying F-isocrystal receives a cohomological interpretation. Nygaard used the dRW theory to give an easier proof of the Rudakov-Shafarevich vanishing theorem on K3-surfaces, and to generalize Ogus-Mazur's theorem on the relation between the Newton - and the Hodge polygon of the crystalline cohomology of a smooth proper scheme. Building on work by Illusie-Raynaud, Ekedahl constructed a duality theory for the dRW complex and gave a new proof for Poincare duality and the Künneth decomposition for crystalline cohomology. He also analyzed the Hodge-Witt cohomology groups, but these groups and in particular their torsion are still very mysterious and not well understood until today.
There are many variants and generalizations of the de-Rham-Witt complex which show up in different contexts. There is a log-version constructed by Hyodo-Kato, a relative version by Langer-Zink, the overconvergent dRW was constructed by Davis-Langer-Zink, the big dRW for arbitrary rings (or schemes) was constructed by Hesselholt-Madsen; they are related to (log/relative)-crystalline- and rigid cohomology, K-theory, motivic cohomology, and topological Hochschild homology, etc.
In this course we will give a detailed construction of the big Witt vectors and the dRW complex over a general ring, following Hesselholt. Then we will focus on the p-typical theory in positive characteristic and explain the main results of Bloch and Deligne-Illusie. As time permits we will discuss some classical and also some of the more recent developments alluded to above.
Prerequisites: For the construction of the big dRW complex only a basic knowledge of commutative algebra and the theory of differentials is needed. Then we will require a solid background in algebraic geometry, as in Hartshorne's book.
|Lecture||Fr 14 - 16||A3/SR130 (Hinterhaus)|