$q$-crystals and $q$-connections
Algebraic Geometry
2020-10-07 v1 Number Theory
Abstract
We study how the category of -connections depends on the choice of coordinates. We exploit Bhatt's and Scholze's -crystalline site, which is based on a coordinate free formulation of -PD structures, in order to relate -crystals and -connections in the -adic setting. This yields a natural equivalence between the the categories of -connections for different choices of coordinates in the -adic setting. The equivalence can be described explicitly in terms of differential operators. In order to obtain a global equivalence, we patch the -adic differential operators to create a global one. The process is not entirely formal, and we are only able to obtain a global equivalence after inverting .
Cite
@article{arxiv.2010.02504,
title = {$q$-crystals and $q$-connections},
author = {Andre Chatzistamatiou},
journal= {arXiv preprint arXiv:2010.02504},
year = {2020}
}