TY - JOUR

T1 - On the magnitude of a finite dimensional algebra

AU - Chuang, Joseph

AU - King, Alastair

AU - Leinster, Tom

PY - 2016/1/7

Y1 - 2016/1/7

N2 - 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 $\chi_A(S,S)$.

AB - 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 $\chi_A(S,S)$.

UR - http://www.tac.mta.ca/tac/volumes/31/3/31-03abs.html

UR - http://www.tac.mta.ca/tac/volumes/31/3/31-03abs.html

M3 - Article

VL - 31

SP - 63

EP - 72

JO - Theory and Applications of Categories

JF - Theory and Applications of Categories

SN - 1201-561X

IS - 3

ER -