Relate synthetic cyclotomic modules to prismatic Dieudonné theory

Establish an equivalence between the graded-zero T_ev-fixed part of the synthetic cyclotomic module induced by the fully faithful embedding D_CycSyn: FBT_A^{op} → CycSyn_{F^{\star}_ev THH(A; Z_p)} and the prismatic Dieudonné cohomology by proving that gr^0(D_CycSyn)^{T_ev} ≃ RΓ(A; D_), where D_ denotes the prismatic Dieudonné functor of Anschütz–Le Bras for every smooth algebra A over a perfect F_p-algebra k, with the additional structure of an 𝒩_A^{\star}-module equipped with the v-adic filtration and Frobenius action preserved.

Background

The paper constructs an ∞-category of cyclotomic synthetic spectra and identifies a Postnikov t-structure whose heart corresponds to derived V-complete η-deformed Cartier complexes. For smooth F_p-algebras A, the authors prove that π_0{cyc,P}(F{\star}_ev THH(A)) recovers the de Rham–Witt complex, yielding a bridge between synthetic constructions and classical cohomology.

They further establish Theorem A, which gives a fully faithful embedding of the opposite category of formal p-divisible groups over A into the ∞-category of modules over F{\star}_ev THH(A; Z_p) in CycSyn. The stated conjecture aims to connect this synthetic embedding D_CycSyn with the prismatic Dieudonné functor by identifying the graded-zero T_ev-fixed piece with RΓ(A; D_), thus aligning the synthetic cyclotomic framework with prismatic Dieudonné theory.

References

Conjecture Let k be a perfect F_p-algebra and let A be a smooth k-algebra. Let D_{CycSyn}: \mathrm{FBT}A{op}\hookrightarrow CycSyn{F{ \star}evTHH(A;Z_p)} be the fully faithful embedding of Theorem~\ref{thm: thmA}. Note that by taking \mathrm{gr}0(-){T_ev} of a F{ \star}_ev THH(A;Z_p)-module in CycSyn we get a \mathcal{N}{ \star} _{A}-module with a filtration given by the v-adic filtration. Additionally the cyclotomic synthetic Frobenius induces a Frobenius map on this module. Then, considered with this extra structure, there is an equivalence \mathrm{gr}0(D{CycSyn}){Tev}\simeq \mathrm{R\Gamma}(A; D_ ) where D_ is the prismatic Dieudonn{ e} functor of .

— Cyclotomic synthetic spectra (2411.19929 - Antieau et al., 29 Nov 2024) in Subsection “Lifts of Dieudonné modules over smooth F_p-algebras,” near the end (Conjecture)