Seminário de Álgebra e Combinatória: Rethinking rank: from linear algebra to additive categories

The concept of a rank function on an additive category with cokernels, or more generally, on one admitting finite weak cokernel resolutions, generalises the classical notion of a Sylvester rank function on a ring, which itself extends familiar invariants such as the dimension of a vector space or the rank of a matrix. Analogous notions have also been developed for triangulated categories and for (d+2)-angulated categories. In this talk, we survey existing notions of rank-type functions and introduce the concepts of pre-rank and rank functions on the morphisms of an additive category, which include all previously studied rank-type invariants. We prove that pre-rank functions on the morphisms of an additive category C correspond bijectively to rank functions on the module category of the opposite of C and we characterise our new class of rank functions via this correspondence. This functorial perspective is crucial for obtaining unique decomposition results, for establishing connections to purity, and for extending theorems and concepts familiar from the case of rank functions on rings to substantially broader settings. The talk is based on joint work in progress with Simone Virili.