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Compact generation of derived categories of stacks

Banff International Research Station for Mathematical Innovation and Discovery

I will give a survey over recent results on derived categories of algebraic stacks with an emphasis on compact generation. In a loose sense, compact objects in the derived category is a replacement for ample line bundles on projective schemes. They also generalize vector bundles but are more flexible. From a different perspective, compact objects are like the coherent sheaves among the quasi-coherent sheaves. I will focus on three questions: (1) When are perfect complexes compact? (2) When is the derived category compactly generated? (3) When are complexes of quasi-coherent sheaves good enough? For schemes, these questions are well understood by work of Thomason, Neeman, Bokstedt, Bondal, Van den Bergh and Lipman. For stacks, the picture is not complete yet but there are satisfactory partial results by Ben-Zvi, Francis, Nadler, Toen, Antieau, Gepner, Lurie, Drinfeld, Gaitsgory, Hall, Neeman and the speaker.

Mathematics; Category theory; homological algebra; Algebraic structures

video/mp4

Author affiliation: Royal Institute of Technology (KTH)

2016-12-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/60056

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/