Twisted differential operators of negative level and prismatic crystals

Abstract

We introduce twisted differential calculus of negative level and prove a descent theorem: Frobenius pullback provides an equivalence between finitely presented modules endowed with a topologically quasi-nilpotent twisted connection of level minus one and those of level zero. We explain how this is related to the existence of a Cartier operator on prismatic crystals. For the sake of readability, we limit ourselves to the case of dimension one.

Prismatic and $q$-crystalline sites of higher level

“…We guess that it would be reasonable to use the category M ∆ (A) and M ∆ (A ′ ) respectively, and so it would be related to our equivalences of the category of crystals with the symbol C ∆ .) So we see that our equivalence of functors fits into the diagram in Proposition 6.9 of [GLSQ20b]. In particular, our argument gives a direct proof of the equivalence α * • ρ * , which they plan to prove indirectly in a forthcoming paper by showing the equivalence of functors G, H. (See Remark 2 after Proposition 6.9 in [GLSQ20b].)…”

“…In Section 5, we explain relations between our result and results in the works of Xu [Xu19], Gros-Le Stum-Quirós [GLSQ20b] and Morrow-Tsuji [MT20]. We will see that our equivalences between the category of crystals on the prismatic site, that on the 1-prismatic site and that on the q-crystalline site fit naturally into the equivalence of Cartier transform in the case modulo p n by Xu, the equivalence between the category of twisted hyper-stratified modules of level −1 and that of level 0 by Gros-Le Stum-Quirós, and a diagram involving the category of crystals on prismatic site, that of generalized representations, that of modules with flat q-Higgs field and that of modules with flat q-connections appearing in the work of Morrow-Tsuji.…”

“…In particular, our argument gives a direct proof of the equivalence α * • ρ * , which they plan to prove indirectly in a forthcoming paper by showing the equivalence of functors G, H. (See Remark 2 after Proposition 6.9 in [GLSQ20b].) Also, our proof of the equivalence α * • ρ * , which is in the style of [Oya17], [Xu19], partially answers 'the hope' in Remark 3 after Proposition 6.9 of [GLSQ20b].…”

“…is the category of twisted hyper-stratified modules of level 0 on A over D (resp. −1 on A ′ over D) defined in Definition 3.9 in [GLSQ20b], G,H and F * are the functors which appear in Proposition 6.9 of [GLSQ20b] and F * is an equivalence of Theorem 4.8 in [GLSQ20b]. (Precisely speaking, it is not clear which category of modules they used in their definition of Strat (0) q (A/D) and Strat (−1) q (A ′ /D).…”

Cartier transform and prismatic crystals

“…This article is the continuation of [GLQ22a] and [GLQ22b]. It is devoted to giving the final arguments realizing our project, first outlined in [Gro20], of putting the local q-twisted Simpson correspondence constructed in [GLQ19] into the perspective of the q-crystalline and prismatic sites theories.…”