Characteristic cycles for coadmissible D-modules on smooth rigid analytic curves

Abstract

Let $\mathfrak{X}$ be a formal smooth curve over a complete discrete valuation ring of mixed characteristic and let ${\mathfrak{X}}_K$ be its generic fiber. We consider respectively over $\mathfrak{X}$ and ${\mathfrak{X}}_K$ the sheaves of differential operators ${\mathcal{D}}_{\mathfrak{X},\infty }$ and Dcap with a rapid convergence condition. In this article, we define a characteristic variety as a subset of the cotangent space $T^{\ast }{\mathfrak{X}}_K$ together with a characteristic cycle for coadmissible Dcap-modules. We deduce a notion of ''sub-holonomicity'' for coadmissible Dcap-modules which is equivalent to being generically an integrable connection. When $\mathfrak{X}$ is quasi-compact, we get an Artinian category of sub-holonomic Dcap-modules which are weakly-holonomic. Moreover, we prove that a coadmissible Dcap-modules is sub-holonomic if and only if the corresponding coadmissible ${\mathcal{D}}_{\mathfrak{X},\infty }$-module is.