Strong equivalence of graded algebras
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909261828718592 |
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| author | Abadie, F. Exel, R. Dokuchaev, M. |
| author_facet | Abadie, F. Exel, R. Dokuchaev, M. |
| contents | 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 are stably isomorphic as graded algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_03513 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong equivalence of graded algebras Abadie, F. Exel, R. Dokuchaev, M. Rings and Algebras 46L55 (Primary) 16W50, 16S35, 16D90 (Secondary) 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 are stably isomorphic as graded algebras. |
| title | Strong equivalence of graded algebras |
| topic | Rings and Algebras 46L55 (Primary) 16W50, 16S35, 16D90 (Secondary) |
| url | https://arxiv.org/abs/2201.03513 |