Crystalline $$\mathbb {Z}_p$$-Representations and $$A_{\inf }$$-Representations with Frobenius

Generalised representations as q-connections in integral $p$-adic Hodge theory

“…Remark 1. 35. We remark that we may also state Theorem 1.34 independently of any choice of framing: there exists a fully faithful embedding…”

“…(iii) Relative Fontaine-Laffaille modules: the second author has shown that Faltings' relative Fontaine-Laffaille modules give rise to objects of Rep µ Γ (A inf (R), ϕ) [35].…”

“…, where the second arrow is the canonical one from §1.1 and the first arrow is the unique W -algebra homomorphism sending T i to [T ♭ i ] (again, see §1.1 for a reminder on the notation) [35,Lem. 64].…”

“…64]. In [35], the second author uses the point of view of generalised representations to re-establish certain aspects of the theory of relative Fontaine-Laffaille modules, notably fully faithfulness of the functor T crys .…”

“…To goal of this paper is to explore and relate different approaches to coefficient systems in relative integral p-adic Hodge theory: the generalised representations of Faltings [18,19], which have been systematically studied by Abbes-Gros-Tsuji [1] and Tsuji [35]; modules with q-connection in the sense of q-de Rham cohomology [32]; and crystals on the prismatic site [10] of Bhatt-Scholze, or equivalently modules with "q-Higgs field" in our context. The equivalence between modules with q-connections and modules with q-Higgs field applies to a range of situations arising in the theory of q-de Rham cohomology and may be viewed as a local, q-deformed Simpson correspondence.…”