J. M. Fontaine (Paris-Sud).

Almost $C_p$-representations and vector bundles
Lecture time: 14:30-16:00

Abstract: Let $K$ be a finite extension of $\mathbb Q_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$.
An $\text{ almost $C_p$-representation of $G_K$}$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb N$, two $G_K$-stable finite dimensional sub-$\mathbb Q_p$-vector spaces $U$ of $V$, $U'$ of $C_p^d$, and a $G_K$-equivariant isomorphism
$$V/U\longrightarrow C_p^d/U'\ .$$
These representations form an abelian category $\mathcal{C}(G_K)$.
The main purpose of this series of lectures is to construct an equivalence of triangulated categories
$$D^b(\mathcal{M}(G_K))\longrightarrow D^b(\mathcal{C}(G_K))$$
and to describe possible generalisations.
Main topics:
-- Construction and main properties of the curve X, generalisations,
-- Classification of coherent $\mathcal{O}_X$-modules,
-- The category $\mathcal{M}(G_K)$,
-- Almost $C_p$-representations,
-- Effective coherent $\mathcal{O}_X[G_K]$-modules and effective almost $C_p$-representations,
-- The main theorem.
-- The étale, the pro-étale and the $v$ sites of an adic space $S$.
-- Possible generalisations.
-- The case where $S$ is the adic space associated to $C_p$. Banach-Colmez spaces.

L. Illusie.

Revisiting the de Rham-Witt complex
Lecture time: 14:30-16:00

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.

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