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Integral $$p$$-adic Hodge theory. (English) Zbl 1446.14011
Authors’ abstract: “We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of $$\mathbb{C}_p.$$ It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known $$p$$-adic cohomology theories, such as crystalline, de Rham and étale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings’ almost purity theorem, along with a certain functor $$L\eta$$ on the derived category, defined previously by Berthelot-Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham-Witt complexes of Langer-Zink, and can be computed as a $$q$$-deformation of de Rham cohomology.”
The convention is that $$\mathbb{C}_p$$ is a completed algebraic closure of a complete discretely valued nonarchimedean extension $$K$$ of $$\mathbb{Q}_p$$. One often writes $$\mathbb{C}_p$$ as $$\mathbb{C}$$, as $$p$$ is understood throughout.
The paper provides a wealth of information on the precise connection with other known $$p$$-adic cohomology theories, and also on the relation of their work to earlier constructions.
An important feature of the new cohomology is the better grip on torsion. For instance, it is used to show that $$\dim_k\mathrm{H}^i_{\mathrm{dR}}(\mathfrak X_k)\geq \dim_{\mathbf F_p}\mathrm{H}^i_{\mathrm{\acute{e}t}}(\mathfrak X_{\mathbf{C}},\mathbf F_p)$$. Here $$\mathfrak X$$ is a proper smooth formal scheme over $$\mathcal O_K$$, where $$\mathcal O_K$$ is the ring of integers in $$K$$ with perfect residue field $$k$$. And $$\mathfrak X_{\mathbf C}$$ is the (geometric) rigid-analytic generic fiber of $$\mathfrak X$$.
Let $$A_{\mathrm{inf}}$$ be the ring of Witt vectors over an inverse limit $$\mathcal O^\flat$$ of iterated Frobenius maps on $$\mathcal O_K/p$$. The ring $$\mathcal O^\flat$$ is called the tilt of $$\mathcal O$$ and comes with a Frobenius map $$\varphi$$. The new cohomology theory is given by a complex $$A\Omega_{\mathfrak X}$$ of sheaves of $$A_{\mathrm{inf}}$$-modules on $${\mathfrak X}_{\mathrm{Zar}}$$. It carries the structure of a commutative $$A_{\mathrm{inf}}$$ algebra in a derived category. (The derived category is not to be viewed as a triangulated category, as the construction of $$A\Omega_{\mathfrak X}$$ involves a non-exact functor.) One has a Frobenius action on $$A\Omega_{\mathfrak X}$$. It does not come from an action on $${\mathfrak X}$$ but from an action on its ‘tilt’.
The construction of $$A\Omega_{\mathfrak X}$$ and the proof of its properties take many steps. The authors explain things carefully. This review cannot do justice to all the material in the paper.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 11G25 Varieties over finite and local fields 14B99 Local theory in algebraic geometry 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14G45 Perfectoid spaces and mixed characteristic
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