Derived categories of representations of small categories over commutative noetherian rings

TL;DR

This paper characterizes the localizing subcategories of derived categories of small categories over noetherian rings, linking them to the ring's spectrum and residue fields, and confirms the telescope conjecture in specific cases.

Contribution

It provides a parametrization of localizing subcategories in terms of the ring spectrum and residue fields, and offers a complete classification for Dynkin quivers over noetherian rings.

Findings

Parametrization of localizing subcategories via ring spectrum and residue fields.

Complete classification of localizing and thick subcategories for Dynkin quivers.

Verification of the telescope conjecture in the context studied.

Abstract

We study the derived categories of small categories over commutative noetherian rings. Our main result is a parametrization of the localizing subcategories in terms of the spectrum of the ring and the localizing subcategories over residue fields. In the special case of representations of Dynkin quivers over a commutative noetherian ring we give a complete description of the localizing subcategories of the derived category, a complete description of the thick subcategories of the perfect complexes and show the telescope conjecture holds. We also present some results concerning the telescope conjecture more generally.