Geometric purity and the frame of smashing ideals

TL;DR

This paper introduces geometric purity in tt-categories, explores its properties and applications, and uses it to address the spatiality of smashing ideals, providing new insights and ruling out previous counterexamples.

Contribution

It defines geometric purity, analyzes its properties, and applies it to the problem of spatiality of smashing ideals in tt-categories.

Findings

Geometric purity is stronger than ordinary purity.

Geometrically pure-injective objects arise from tt-stalks.

The approach rules out known counterexamples to spatiality.

Abstract

We introduce the notion of geometric purity in rigidly-compactly generated tt-categories by considering exact triangles that are pure at each tt-stalk. We develop a systematic study of this concept, including examples and applications. In particular, we show that geometric purity is, in general, strictly stronger than ordinary purity, and that it naturally leads to the notion of geometrically pure-injective objects. We prove that such objects arise as pushforwards of pure-injective objects from suitable tt-stalks. Moreover, we give a detailed analysis of indecomposable geometrically pure-injective objects in the derived category of the projective line. Under mild additional assumptions, we identify the geometric part of the Ziegler spectrum as a closed subset. As an application, we demonstrate that this new notion of purity can be used to tackle the problem of spatiality of the frame of smashing ideals via the geometric Ziegler spectrum. In particular, we show that our approach rules out the counterexamples of Balchin and Stevenson to existing methods.