$q$-de Rham complexes of higher level

Abstract: 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-Quir\'os. The complexes we construct can also be regarded as the higher-level analogs of the $q$-de Rham complex.