Deformations of overconvergent isocrystals on the projective line

TL;DR

This paper studies the formal deformation theory of overconvergent isocrystals on the projective line over a perfect field, focusing on fixed local monodromy and connecting Hochschild cochains to de Rham complexes.

Contribution

It establishes new results on deformations of overconvergent isocrystals with fixed local monodromy and links Hochschild cochains to de Rham complexes for differential modules.

Findings

Hochschild cochain complex governs module deformations over associative algebras.

Deformation theory of overconvergent isocrystals with fixed local monodromy is characterized.

Connection between Hochschild cochains and de Rham complexes enhances understanding of differential modules.

Abstract

Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on $U$ with fixed "local monodromy" along $Z$. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring.