The local-to-global principle for triangulated categories via dimension functions

TL;DR

This paper introduces a general criterion based on dimension functions to verify the local-to-global principle in tensor triangulated categories, providing new proofs and classifications in algebraic settings.

Contribution

It develops an abstract, inductive criterion for the local-to-global principle and applies it to classify subcategories in derived categories of specific rings.

Findings

Proves the local-to-global principle for categories with a model and noetherian spectrum.

Provides new conditions on spectra ensuring the principle holds.

Classifies localising subcategories in derived categories of semi-artinian absolutely flat rings.

Abstract

We formulate a general abstract criterion for verifying the local-to-global principle for a rigidly-compactly generated tensor triangulated category. Our approach is based upon an inductive construction using dimension functions. Using our criterion we give a new proof of the theorem that the local-to-global principle holds for such categories when they have a model and the spectrum of the compacts is noetherian. As further applications we give a new set of conditions on the spectrum of the compacts that guarantee the local-to-global principle holds and use this to classify localising subcategories in the derived category of a semi-artinian absolutely flat ring.