Signed Euler--Smith Profiles of Graded Algebras

Abstract

Matrix Hilbert series retain local elementary-divisor data that their determinants discard. For locally finite positively graded elementary algebras whose vertex simples are of type $F{P}_{\infty }$ and whose matrix Hilbert series is rational, we construct from possibly infinite minimal resolutions a rational Euler matrix and its signed local Smith profile at $x=1$. The profile refines pole-order growth for perfect modules and is invariant under shift-compatible graded Morita and perfect-derived equivalences. Every finite integer profile occurs, while a wholly negative profile forces exponential corner growth. This extends the nonnegative local Smith framework for twisted Calabi--Yau algebras to a setting in which the Euler matrix may itself have poles.