Motivated by the work of Cohn and Schofield on Sylvester rank functions on rings, Chuang and Lazarev have recently introduced the notion of a rank function on a triangulated category. It turns out that a rank function on a category C can be recast as translation-invariant additive function on its abelianisation mod C. As a consequence, integral rank functions have a unique decomposition into irreducible ones, and they are related to a number of important concepts associated to the localisation theory of mod C. When C is the subcategory of compact objects of a compactly generated triangulated category T, these connections become particularly nice and provide a link between rank functions on C and smashing localisations of T. In particular, when C is the perfect derived category per(A) of a DG algebra A, this allows us to classify homological epimorphisms from A to B with per(B) locally finite via special rank functions, extending a result of Chuang and Lazarev.
This talk is based on joint work with Mikhail Gorsky, Frederik Marks and Alexandra Zvonareva.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
