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 $\chi_A(S,S)$.
Keywords: algebra, magnitude, indecomposable projective, simple module, Cartan matrix, Euler form, Cartan determinant conjecture
2010 MSC: 18G99 (primary), 16D40, 16D60, 16G99, 18E05, 18G15
Theory and Applications of Categories, Vol. 31, 2016, No. 3, pp 63-72.
Published 2016-01-07.
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