In the category of modules over a ring, purity may be viewed as a weakening of splitting – a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. However, in general, it can be hard to check whether a class is closed under purity operations. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and illustrate this with a range of examples, often from the tensor-triangular perspective. I will also discuss an application to the study of definable subcategories of triangulated categories.
This is joint work with Isaac Bird.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
