Derived categories of representations of small categories over commutative noetherian rings

Benjamin Antieau and Greg Stevenson

Vol. 283 (2016), No. 1, 21–42

DOI: 10.2140/pjm.2016.283.21

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 and a complete description of the thick subcategories of the perfect complexes. We also show that the telescope conjecture holds in this setting and we present some results concerning the telescope conjecture more generally.

Keywords

derived categories, localizations, telescope conjecture

Mathematical Subject Classification 2010

Primary: 16E35, 16G20

Secondary: 13D09, 18G55

Milestones

Received: 3 July 2015

Revised: 28 October 2015

Accepted: 5 November 2015

Published: 14 June 2016