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 o...

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Bibliographic Details
Published inarXiv.org
Main Authors Abadie, F, Exel, R, Dokuchaev, M
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.07.2024
Subjects
Algebra
Equivalence
Globalization
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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 \(\alpha \) is globalizable up to Morita equivalence; if such a globalization \(\beta \) is minimal, then the skew group algebras by \(\alpha \) and \(\beta \) are graded-equivalent; moreover, \(\beta \) is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras are stably isomorphic as graded algebras.
ISSN:2331-8422