Notes on generalizations of local Ogus-Vologodsky correspondence

Abstract

Given a smooth scheme over Z/p n Z with a lift of relative Frobenius to Z/p n+1 Z, we construct a functor from the category of Higgs modules to that of modules with integrable connections as the composite of the level raising inverse image functors from the category of modules with integrable p mconnections to that of modules with integrable p m−1 -connections for 1 ≤ m ≤ n. In the case m = 1, we prove that the level raising inverse image functor is an equivalence when restricted to quasi-nilpotent objects, which generalizes a local result of Ogus-Vologodsky. We also prove that the above level raising inverse image functor for a smooth p-adic formal scheme induces an equivalence of Q-linearized categories for general m when restricted to nilpotent objects (in strong sense), under a strong condition on Frobenius lift. We also prove a similar result for the category of modules with integrable p m -Wittconnections.

Prismatic and $q$-crystalline sites of higher level

“…It is easy to see that the functors β 0 , β 1 are continuous and so the functors above induce the commutative diagram: in Lemma 4.14. The functor ϕ is constructed in Proposition 2.5 of [Shi12] and is an equivalence. By composing the functors that are equivalences, we obtain an equivalence ̟ from the category of crystals on the prismatic site to the category of modules with integrable p-connection.…”

Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups

“…More details about the category will be given below. In comparison with the construction of Shiho [29], one finds that the existence of a Frobenius lifting over a chosen lifting of X ′ over W n+1 is not assumed in our construction. On the other hand, we do not know whether the functor C −1 n is fully faithful for a proper W n -scheme when n ≥ 2.…”

“…An anonymous referee has kindly pointed to us that the work of A. Shiho [29] is related to our construction below. Recall, for each n ∈ N, S n = Spec W n and F Sn : S n → S n the Frobenius automorphism.…”

“…Recall, for each n ∈ N, S n = Spec W n and F Sn : S n → S n the Frobenius automorphism. In [29] Shiho constructs a functor from the category of quasi-nilpotent Higgs modules on X (n) n to the category of quasi-nilpotent flat modules on X n , where X n is a smooth scheme over W n and X…”

“…n -modules with integrable p-connections of Shiho [29] which we shall explain later. The motivation to introduce this new category is mainly because of the necessity to make sense of p-powers in the denominators appearing in the Taylor formula (4.16.1).…”