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arXiv · 2201.03513

Strong equivalence of graded algebras

F. Abadie·R. Exel·M. Dokuchaev
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Abstract

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.

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F. Abadie, R. Exel, M. Dokuchaev. 2022-01-10. Strong equivalence of graded algebras. https://arxiv.org/abs/2201.03513

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