Derived categories of absolutely flat rings

TL;DR

This paper investigates the structure of derived categories of absolutely flat rings, establishing a classification of localising subcategories via spectrum subsets and confirming the telescope conjecture for these categories.

Contribution

It provides a classification of localising subcategories of D(R) using spectrum subsets and proves the telescope conjecture for derived categories of absolutely flat rings.

Findings

Subsets of the spectrum parametrize localising subcategories.

The telescope conjecture holds for D(R).

An example of a cohomological Bousfield class not being a Bousfield class is provided.

Abstract

Let S be a commutative ring with topologically noetherian spectrum and let R be the absolutely flat approximation of S. We prove that subsets of the spectrum of R parametrise the localising subcategories of D(R). Moreover, we prove the telescope conjecture holds for D(R). We also consider unbounded derived categories of absolutely flat rings which are not semi-artinian and exhibit an example of a cohomological Bousfield class that is not a Bousfield class.