Unstable arithmetic fracture squares in $\infty$-topoi

TL;DR

This paper demonstrates that in many $ abla$-topoi, unstable arithmetic fracture squares can reconstruct nilpotent sheaves from their rational and p-completions, extending classical fracture square concepts to an $ abla$-topos setting.

Contribution

It introduces the existence of unstable arithmetic fracture squares in a broad class of $ abla$-topoi, generalizing the classical stable fracture square framework.

Findings

Existence of unstable arithmetic fracture squares in many $ abla$-topoi.

Reconstruction of nilpotent sheaves from rational and p-completions.

Extension of fracture square theory to unstable and $ abla$-topos contexts.

Abstract

We show that for a large class of $\infty$-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf $F$ as the pullback of the rationalization of $F$ with the product of the $p$-completions of $F$ ranging over all primes $p\in\mathbb Z$.