Magnitude of module categories and special biserial algebras

Comment

We prove the conjecture of Børve, Horiatakis, and Kalck that the magnitude of the module category of a representation-finite algebra over an algebraically closed field is at least the number of simple modules, with equality precisely for special biserial algebras.

For representation-directed algebras, we relate magnitude to Serre subcategories and ideal quotients, and compute the magnitude of the resulting categories of poset representations. The general case reduces to the representation-directed case through algebras obtained by grading a standard form.

The paper is available on arXiv.