Syntomification and crystalline local systems

Abstract

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf}{\mathcal{O}}_K$ where $K/{\mathbf{Q}}_p$ is a finite extension. We show that reflexive sheaves on the stack ${\mathrm{X}}^{\mathrm{Syn}}$ are equivalent to ${\mathbf{Z}}_p$-lattices in crystalline local systems on the rigid generic fiber ${\mathrm{X}}_{\hat{I}}\cdot$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on ${\mathrm{X}}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf}{\mathcal{O}}_K$ is smooth and proper that $\mathsf{Perf}({\mathrm{X}}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.

Revisions/corrections to the end of section 2 and 3.1.1. Minor changes elsewhere. Comments welcome!