Support and vanishing for non-Noetherian rings and tensor triangulated categories

TL;DR

This paper introduces a new support theory for complexes over non-Noetherian rings and tensor triangulated categories, establishing vanishing theorems and exploring the topology of the Balmer spectrum.

Contribution

It defines and characterizes small support in a broad setting, connecting support with topology and lattice structures in tensor triangulated categories.

Findings

Support detects vanishing in certain contexts.

Support theory extends to non-Noetherian rings and tensor categories.

Relations between support, Balmer spectrum topology, and Bousfield lattice are established.

Abstract

We define and characterise small support for complexes over non-Noetherian rings and in this context prove a vanishing theorem for modules. Our definition of support makes sense for any rigidly compactly generated tensor triangulated category. Working in this generality, we establish basic properties of support and investigate when it detects vanishing. We use pointless topology to relate support, the topology of the Balmer spectrum, and the structure of the idempotent Bousfield lattice.