# Minicorso di Geometria Algebrica e Teoria dei Numeri: “Revisiting the de Rham-Witt complex”

## 10 - 30 Novembre 2017, ore 14:30 - L. Illusie

Minicorso di Geometria Algebrica e Teoria dei Numeri: “Revisiting the de Rham-Witt complex”

Giovedì 16 Novembre - Aula 2BC30
Venerdì 17 Novembre - Aula 2AB45
Giovedì 23 Novembre - Aula 2BC30
Venerdì 24 Novembre - Aula 2AB45
Giovedì 30 Novembre - Aula 2BC30

Abstract
The de Rham-Witt complex was constructed by Spencer Bloch in the mid 1970's as a tool to analyze the crystalline cohomology of proper smooth schemes over a perfect field of characteristic $p >0$, with its action of Frobenius, and describe its relations with other types of cohomology, like Hodge cohomology or Serre's Witt vector cohomology. Since then many developments have occurred.
After recalling the history of the subject, I will explain the main construction and in the case of a polynomial algebra give its simple description by the so-called complex of integral forms. I will then describe the local structure of the de Rham-Witt complex for smooth schemes over a perfect field and its application to the calculation of crystalline cohomology. In the proper smooth case, I will discuss the slope spectral sequence and the main finiteness properties of the cohomology of the de Rham-Witt complex in terms of coherent complexes over the Raynaud ring. I will mention a few complements (logarithmic Hodge-Witt sheaves, Hyodo-Kato log de Rham-Witt complex, Langer-Zink relative variants), and make a tentative list of open problems.