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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 FP∞ 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.
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BibTeX
@misc{kaygun2026signed,
title = {{Signed Euler--Smith Profiles of Graded Algebras}},
author = {Atabey Kaygun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35248},
eprint = {2609.35248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35248},
}
RIS
TY - UNPB
TI - Signed Euler--Smith Profiles of Graded Algebras
AU - Atabey Kaygun
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35248
UR - https://arxiv.org/abs/2609.35248
AB - 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 FP_∞ 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.
ER -