Conservative geometric functors via purity

TL;DR

This paper introduces a criterion based on purity for when a family of geometric functors is jointly conservative, with applications to limits of ring spectra.

Contribution

It develops the concept of pure descendability and applies it to analyze geometric functors in triangulated categories.

Findings

Established a purity-based criterion for joint conservativity.

Introduced the notion of pure descendability.

Applied the criterion to limits of ring spectra.

Abstract

We establish a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra.