We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of \(\mathbf {C}_{p}\). 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.

我们为\（\ mathbf {C} _ {p} \）的整数环上的适当平滑（形式）方案构造了新的同调理论。它采用Dieudonné模块的混合特征类似物中的值，Dieudonné模块先前被Fargues定义为Breuil–Kisin模块的一种版本。值得注意的是，该同调理论专门用于所有其他已知的\（p \）- adic同调理论，例如晶体，de Rham和étale同调论，这使我们能够证明强大的积分比较定理。

谐函数理论的构建依赖于Faltings的几乎纯定理，以及衍生类别上的某个函子\（L \ eta \），这是Berthelot-Ogus先前定义的。在仿射片断上，我们的同调理论承认与Langer-Zink的de Rham-Witt络合物理论有关，并且可以计算为de Rham同调的\（q \）-变形。