On triangulated categories with metrics

Abstract

Neeman has recently initiated the use of metrics, and approximations via metrics, to study triangulated categories. In this thesis, we use these techniques to prove results which have application in algebraic geometry. First of all we define a generalisation of the notion of approximable triangulated categories. We prove some Brown representability type theorems for the compact objects of such categories. Further, we show that for nice schemes, the homotopy category of injectives satisfies the conditions of this new definition, which gives us Brown representability type theorems for the corresponding bounded derived category of coherent sheaves. The second major application is to the construction of new semiorthogonal decompositions from gives ones. This generalises recent work by Kuznetsov, Shinder, and Bondarko. Finally, we discuss joint work on bounded t-structures and the finitistic dimension of a triangulated category, which generalises a powerful new theorem by Neeman.