Change of weights operations for triangulated $({\varphi},{\Gamma})$-modules

TL;DR

This paper explores the change of weights operation on triangulated $(\varphi,\Gamma)$-modules, interpreting it as pullbacks, and demonstrates its compatibility with translation functors in the non-critical crystabelline case.

Contribution

It provides a geometric interpretation of the change of weights operation as pullbacks and establishes its compatibility with translation functors in specific cases.

Findings

Interpretation of change of weights as pullbacks in the trianguline case

Proof of compatibility with translation functors in non-critical crystabelline case

Connection with stacks and 1-1 correspondence by Yiwen Ding

Abstract

Zhixiang Wu has shown the existence of "change of weights" operation on $(\varphi,\Gamma)$-modules in families (arxiv:2405.16637). We interpret it in the trianguline case as pullbacks with a discussion on related stacks. Finally, we prove that it intertwines well with translation functors via a 1-1 correspondence defined by Yiwen Ding (arXiv:2407.21237) in the non-critical crystabelline case.