Twisted differential operators and $q$-crystals
Michel Gros, Bernard Le Stum, Adolfo Quir\'os
TL;DR
This paper explores the theory of q-crystals and their relation to twisted differential operators, providing a new perspective on q-crystalline cohomology and module structures in a simplified one-dimensional setting.
Contribution
It introduces a novel interpretation of q-crystals as modules with stratification and connects them to twisted differential operators, expanding the understanding of q-crystalline cohomology.
Findings
Relation between q-PD-envelopes and divided polynomial twisted algebras
Interpretation of q-crystals as modules with stratification
Association of modules on twisted differential operator rings to q-crystals
Abstract
We discuss the notion of a q-PD-envelope considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted algebra. Together with an interpretation of crystals on the q-crystalline site, that we call q-crystals, as modules endowed with some kind of stratification, it allows us to associate a module on the ring of twisted differential operators to any q-crystal. For simplicity, we explain here only the one dimensional case.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics