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.

S. Bloch, Algebraic K-theory and crystalline cohomology.

Publ. Math., Inst. Hautes Étud. Sci. 47, 187-268 (1977).

L. Illusie, Complexe de De Rham-Witt et cohomologie cristalline.

Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979).

W. E. Lang; N. O. Nygaard, A short proof of the Rydakov-Safarevic theorem.

Math. Ann. 251, 171-173 (1980).

N. O. Nygaard, Slopes of powers of Frobenius on crystalline cohomology.

Ann. Sci. Éc. Norm. Supér. (4) 14, 369-401 (1981).

L. Illusie; M. Raynaud, Les suites spectrales associées au complexe de De Rham-Witt.

Publ. Math., Inst. Hautes Étud. Sci. 57, 73-212 (1983).

L. Illusie, Finiteness, duality, and Künneth theorems in the cohomology of the De Rham Witt complex.

Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 20-72 (1983).

T. Ekedahl, On the multiplicative properties of the de Rham-Witt complex. I.

Ark. Mat. 22, 185-239 (1984).

T. Ekedahl, On the multiplicative properties of the de Rham-Witt complex. II.

Ark. Mat. 23, 53-102 (1985).

M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique.

Mém. Soc. Math. Fr., Nouv. Sér. 21, 87 p. (1985).

T. Geisser; M. Levine, The K-theory of fields in characteristic p.

Invent. Math. 139, No.3, 459-493 (2000).

A. Langer; T. Zink, De Rham-Witt cohomology for a proper and smooth morphism.

J. Inst. Math. Jussieu 3, No. 2, 231-314 (2004).

S. Bloch, Crystals and de Rham-Witt connections.

J. Inst. Math. Jussieu 3, No. 3, 315-326 (2004).

L. Hesselholt, I. Madsen, On the de Rham-Witt complex in mixed characteristic.

Ann. Sci. Ec. Norm. Sup. 37 (4), 1-43 (2004).

C. Davis; A. Langer; T. Zink, Overconvergent de Rham-Witt cohomology.

Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 197-262 (2011).

L. Hesselholt, The big de Rham-Witt complex.

Acta Math. 214, 135-207 (2015).

J. Cuntz; C. Deninger, Witt vector rings and the relative de Rham Witt complex.

J. Algebra 440, 545-593 (2015).

Prerequisits: 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.