Strong equivalence of graded algebras

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. As to a more general idempotent graded algebra B, we point out that the...

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Bibliographic Details
Published inJournal of algebra Vol. 659; pp. 818 - 858
Main Authors Abadie, F., Dokuchaev, M., Exel, R.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2024
Subjects
Graded algebra
Morita equivalence
Partial action
Skew group ring
Smash product
secondary
Partial action
Morita equivalence
Graded algebra
Skew group ring
primary
Smash product
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Summary:We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. As to a more general idempotent graded algebra B, we point out that the Cohen-Montgomery duality holds for B, and B is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action α is globalizable up to Morita equivalence; if such a globalization β is minimal, then the skew group algebras by α and β are graded-equivalent; moreover, β is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras with orthogonal local units are stably isomorphic as graded algebras.
ISSN:0021-8693
DOI:10.1016/j.jalgebra.2024.07.014