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

Abstract

We relate various approaches to coefficient systems in relative integral p-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over A inf inspired by Faltings, modules with q-connection in the sense of q-de Rham cohomology, crystals on the prismatic site of Bhatt-Scholze, and q-deformations of Higgs bundles.

On the Hodge--Tate crystals over O_K

“…On the other hand, Theorem 4.4 shows that the existence of the Frobenius structure ensures the condition (3) in Definition 5.1, especially the pseudo-nilpotency condition. This is similar to the phenomenon appearing in [MT20]. Conversely, if one can prove that the Frobenius structure implies the condition (3) in Definition 5.1, Theorem 5.2 then follows.…”

“…A posteriori, the existence of the Frobenius ensures the "nilpotency" of the τ -connection after [BS21] and [DL21]. This is similar to the phenomenon appearing in [MT20]. We can show this is true in the three cases above, i.e.…”

“…This category will be shown to be equivalent to the category Vect((O K ) ∆ , O ∆ ). The idea of using the τ -connection is inspired by [MT20].…”

“…Proof. The proof is essentially appears in the proof of [MT20,Proposition 3.18]. By construction, the base change of ǫ along the multiplication map ∆ : S 1 → S is the identity morphism.…”

“…One application of our previous results on the Hodge-Tate crystals is to show the "nilpotency" of the τ -connection on the crystalline Breuil-Kisin modules. The idea of considering the τ -connection is inspired by [MT20].…”