PhD Defense Rune Harder Bak

Title: Tensor abelian categories - in a non-commutative setting

Tensor abelian categories provide a framework for studying both the additive (abelian) and the multiplicative (monoidal) structure of categories like abelian groups, modules over rings, chain complexes, (differential) graded modules, quasi-coherent sheaves and functor categories, even in the non-commutative setting.

In the first paper, we prove in this framework a classic theorem of Lazard and Govorov which states that flat modules are precisely the direct limit closure of the finitely generated projective modules. The general result reproves this and other ad hoc examples and provide new results in other categories including the category of differential graded modules. In the second paper we study quiver representations in such categories and characterize various classes of representations.

This again generalizes characterizations in the categories of modules over rings, but provides new insight even in this case.

In the last paper we study a generalization of the prime ideal spectrum in this setting, namely the atom spectrum. This has many good theoretical properties but concrete calculations are few. We provide a method for calculating this with several concrete examples.

Supervisor:  Ass. Prof. Henrik G. Holm Math, University of Copenhagen 

Assessment committee:
Prof. Prof. Ian Kiming (Chairman), MATH, University of Copenhagen
Prof. Lars Winther Christensen, Texas Tech University
Ass. Prof. Dolors Herbera, Univerität Autonoma de Barcelona