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arXiv · 2608.14057

Local Smith Profiles of Twisted Calabi--Yau Algebras

Atabey Kaygun
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Abstract

The matrix Hilbert series of a locally finite elementary twisted Calabi--Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand--Kirillov dimension, which records only the largest of them. We show that the Calabi--Yau symmetry makes this local data rigid: the algebra decomposes into ring factors according to the average Artin--Schelter index along Nakayama cycles, and after a local normalization the symmetry induces a nonsingular linking form on the Smith cokernel together with a finite-order Nakayama action on its layers, forcing reciprocal-eigenvalue and parity constraints on the multiplicities of the exponents. We compute the complete local data for cyclic skew-group algebras, derive a parity sieve for quiver classifications in dimension three, and realize, as an iterated smash product of a graded down-up algebra, a four-vertex type that a recent classification had left open.

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Atabey Kaygun. 2026-08-14. Local Smith Profiles of Twisted Calabi--Yau Algebras. https://arxiv.org/abs/2608.14057

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