Towards affinoid Duflo Theorem I: Twisted differential operators

TL;DR

This paper develops a theory of deformed Picard algebroids and twisted differential operators over schemes, establishing a bijection between them and extending classical notions to a more general algebraic setting.

Contribution

It introduces deformed Picard algebroids and twisted differential operators, proving their natural bijection and defining pullbacks that generalize classical concepts.

Findings

Established a bijection between deformed Picard algebroids and twisted differential operators.

Defined pullback operations for twisted differential operators extending classical cases.

Proved descent theorems for modules over twisted differential operators under torsors.

Abstract

For a commutative ring $R$, we define the notions of deformed Picard algebroids and deformed twisted differential operators on a smooth, separated, locally of finite type $R$-scheme and prove these are in a natural bijection. We then define the pullback of a sheaf of twisted differential operators that reduces to the classical definition when $R=\mathbb{C}$. Finally, for modules over twisted differential operators, we prove a theorem for the descent under a locally trivial torsor.