The Relative (de-)Perfectoidification Functor and Motivic p-Adic Cohomologies

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“…Proof. Since the "underlining" functor is left exact, it is enough to prove that if f : M ′ → M is a surjective map between two Banach A-modules, the map f : M ′ → M remains surjective; in other words, that whenever S is an extremally disconnected set and g : S → M is a continuous map, there is a continuous map g ′ : S → M ′ lifting g. But the image g(S) is compact, and thus by [Trèves 1967 (1) There is a canonical equivalence g * R dR (X/S)…”

The de Rham–Fargues–Fontaine cohomology

“…In particular, we deduce that it is motivic, i.e., it can be defined as a contravariant realization functor dR S : RigDA(S) → QCoh(S) op on the (unbounded, derived, stable, étale) category RigDA(S) of rigid analytic motives over S with values in the infinity-category of solid quasicoherent O S -modules. As a matter of fact, in order to prove the properties above we make extensive use of the theory of motives, and more specifically of their six-functor formalism [Ayoub et al 2022] and of a homotopy-based relative version of Artin's approximation lemma (Theorem 3.9) inspired by the absolute motivic proofs given in [Vezzani 2018]. If X is a proper smooth rigid variety over S, dR S (X ) is a perfect complex, whose cohomology groups are vector bundles.…”

“…On the other hand, if P is a characteristic p perfectoid space, one can postcompose it with specialization along a chosen untilt P ♯ → X (P) and get a perfect complex of O P ♯ -modules. By doing so when P = C is an algebraically closed perfectoid field of characteristic p, we recover a construction from [Vezzani 2019b] and also Bhatt, Morrow and Scholze's B + dR (C ♯ )-cohomology [Bhatt et al 2018, Section 13] for each untilt C ♯ of C. This proves that dR FF satisfies all the requirements of conjecture 6.4 in [Scholze 2018]. There is also a connection to rigid cohomology that we sketch at the end of the article.…”

“…Step 3: We can now use the results of [Vezzani 2019a] which do not use the hypothesis K + = K • to conclude. Assume M to be the motive of a variety X which is étale over the some affine space over W (K + ).…”