Kimihiko Li | Searcharxiv

arXiv2024

$q$-de Rham complexes of higher level

In this article, we construct two kinds of de Rham-like complexes which compute the cohomology of complete crystals on the higher-level $q$-crystalline site, which was introduced in a previous article of the author. One complex is the $q$-analog of the higher de Rham complex constructed by Miyatani, and another complex is the $q$-analog of the jet complex constructed by Le Stum-Quirós. The complexes we construct can also be regarded as the higher-level analogs of the $q$-de Rham complex.

math.AG↗

arXiv2022

Prismatic and $q$-crystalline sites of higher level

In this article, we define the $m$-prismatic site and the $m$-$q$-crystalline site, which are higher level analogs of the prismatic site and the $q$-crystalline site respectively. We prove a certain equivalence between the category of crystals on the $m$-prismatic site (resp. the $m$-$q$-crystalline site) and that on the prismatic site (resp. the $q$-crystalline site), which can be regarded as the prismatic (resp. the $q$-crystalline) analog of the Frobenius descent due to Berthelot and the Cartier transform due to Ogus-Vologodsky, Oyama and Xu. We also prove the equivalence between the category of crystals on the $m$-prismatic site and that on the $(m-1)$-$q$-crystalline site.

math.AG↗