On the magnitude of a finite dimensional algebra

TL;DR

This paper explores the concept of magnitude in enriched categories, demonstrating its relevance in algebra by linking it to a known homological invariant of associative algebras, specifically the Euler form.

Contribution

It establishes that the magnitude of the category of indecomposable projective modules equals the Euler form of the algebra, connecting categorical magnitude to algebraic invariants.

Findings

Magnitude of the module category equals the Euler form chi_A(S, S).

Links categorical magnitude to classical algebraic invariants.

Provides a new perspective on algebraic invariants via enriched category theory.

Abstract

There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension. Here we establish its significance in an algebraic context. Specifically, in the representation theory of an associative algebra A, a central role is played by the indecomposable projective A-modules, which form a category enriched in vector spaces. We show that the magnitude of that category is a known homological invariant of the algebra: writing chi_A for the Euler form of A and S for the direct sum of the simple A-modules, it is chi_A(S, S).