On the magnitude of a finite dimensional algebra

Joseph Chuang, Alastair King, Tom Leinster

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

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 $\chi_A(S,S)$.

Original languageEnglish
Pages (from-to)63-72
JournalTheory and Applications of Categories
Volume31
Issue number3
Publication statusPublished - 7 Jan 2016

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