Vol. 283, No. 1, 2016

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ISSN: 0030-8730
Derived categories of representations of small categories over commutative noetherian rings

Benjamin Antieau and Greg Stevenson

Vol. 283 (2016), No. 1, 21–42

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.

derived categories, localizations, telescope conjecture
Mathematical Subject Classification 2010
Primary: 16E35, 16G20
Secondary: 13D09, 18G55
Received: 3 July 2015
Revised: 28 October 2015
Accepted: 5 November 2015
Published: 14 June 2016
Benjamin Antieau
Department of Mathematics, Statistics, and Computer Science
University of Illinois at Chicago
851 S. Morgan St.
Chicago, IL 60607
United States
Greg Stevenson
Faculty of Mathematics
Universität Bielefeld
Universitätstraße 25
D-33615 Bielefeld