Magnitude of module categories

We define an invariant of the module category of a representation-finite algebra by the magnitude of its Auslander algebra.

This invariant will be called the magnitude of the module category.

For bound path algebras, it can be computed as the Euler characteristic of the Auslander--Reiten quiver, in a suitable sense.

To aid the computation of our invariant, we define the Auslander--Reiten--Euler characteristic of a translation quiver.

We build on classical results in Auslander--Reiten theory to determine the magnitude of module categories of biserial algebras, hereditary path algebras, radical square zero bound path algebras, and self-injective bound path algebras.

In these cases, we express our invariant in terms of other known quantities, notably the rank of the Grothendieck group and Coxeter numbers of Dynkin quivers.

Based on our calculations and results, we obtain a conjectural characterisation of representation-finite biserial algebras in terms of the magnitude of the module category and the rank of the Grothendieck group.