The de Rham-Witt complex

Course at FU Berlin, Winter Term 18/19

Kay Rülling



Content

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.

Data

Time Room
Lecture Fr 14 - 16 A3/SR130 (Hinterhaus)

The first lecture will take place on Friday, October 19, 2018.

Literature

  1. S. Bloch, Algebraic K-theory and crystalline cohomology.
    Publ. Math., Inst. Hautes Étud. Sci. 47, 187-268 (1977).
  2. S. Bloch, Crystals and de Rham-Witt connections.
    J. Inst. Math. Jussieu 3, No. 3, 315-326 (2004).
  3. J. Cuntz; C. Deninger, Witt vector rings and the relative de Rham Witt complex.
    J. Algebra 440, 545-593 (2015).
  4. C. Davis; A. Langer; T. Zink, Overconvergent de Rham-Witt cohomology.
    Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 197-262 (2011).
  5. T. Ekedahl, On the multiplicative properties of the de Rham-Witt complex. I.
    Ark. Mat. 22, 185-239 (1984).
  6. T. Ekedahl, On the multiplicative properties of the de Rham-Witt complex. II.
    Ark. Mat. 23, 53-102 (1985).
  7. T. Geisser; M. Levine, The K-theory of fields in characteristic p.
    Invent. Math. 139, No.3, 459-493 (2000).
  8. 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).
  9. L. Hesselholt, I. Madsen, On the de Rham-Witt complex in mixed characteristic.
    Ann. Sci. Ec. Norm. Sup. 37 (4), 1-43 (2004).
  10. L. Hesselholt, The big de Rham-Witt complex .
    Acta Math. 214, 135-207 (2015).
  11. L. Illusie, Complexe de De Rham-Witt et cohomologie cristalline.
    Ann. Sci. Éc. Norm. Supér. (4) 12, 501--661 (1979).
  12. L. Illusie; M. Raynaud, Les suites spectrales associées au complexe de De Rham-Witt.
    Publ. Math., Inst. Hautes Étud. Sci. 57, 73-212 (1983).
  13. 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).
  14. W. E. Lang; N. O. Nygaard, A short proof of the Rydakov-Safarevic theorem.
    Math. Ann. 251, 171-173 (1980).
  15. A. Langer; T. Zink, De Rham-Witt cohomology for a proper and smooth morphism.
    J. Inst. Math. Jussieu 3, No. 2, 231-314 (2004).
  16. N. O. Nygaard, Slopes of powers of Frobenius on crystalline cohomology.
    Ann. Sci. Éc. Norm. Supér. (4) 14, 369-401 (1981).