DG-Semiprimary DG-Algebras, Acyclicity and Hopkins-Levitzki Theorem for DG-Algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917969746984960 |
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| author | Zimmermann, Alexander |
| author_facet | Zimmermann, Alexander |
| contents | We study the analogue of the Hopkins-Levitzky Theorem for dg-algebras $(A,d)$. We first consider the Hopkins approach. Here we show that for acyclic dg-algebras with graded-Artinian algebras of cycles $\ker(d)$, we also have that $(A,d)$ is left dg-Noetherian, and we show that acyclic dg-Artinian dg-algebras are dg-Noetherian. Then, studying the Levitzki approach, we consider a definition of a dg-semiprimary algebra. For dg-semiprimary dg-artinian dg-algebras $(A,d)$, we show that all dg-simple dg-modules are acyclic, and so are all dg-modules with finite dg-composition length. We finally show that dg-Artinian dg-semiprimary dg-algebras with nilpotent dg-radical $dgrad_2(A,d)$ are dg-Noetherian and acyclic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | DG-Semiprimary DG-Algebras, Acyclicity and Hopkins-Levitzki Theorem for DG-Algebras Zimmermann, Alexander Rings and Algebras Representation Theory 16E45 We study the analogue of the Hopkins-Levitzky Theorem for dg-algebras $(A,d)$. We first consider the Hopkins approach. Here we show that for acyclic dg-algebras with graded-Artinian algebras of cycles $\ker(d)$, we also have that $(A,d)$ is left dg-Noetherian, and we show that acyclic dg-Artinian dg-algebras are dg-Noetherian. Then, studying the Levitzki approach, we consider a definition of a dg-semiprimary algebra. For dg-semiprimary dg-artinian dg-algebras $(A,d)$, we show that all dg-simple dg-modules are acyclic, and so are all dg-modules with finite dg-composition length. We finally show that dg-Artinian dg-semiprimary dg-algebras with nilpotent dg-radical $dgrad_2(A,d)$ are dg-Noetherian and acyclic. |
| title | DG-Semiprimary DG-Algebras, Acyclicity and Hopkins-Levitzki Theorem for DG-Algebras |
| topic | Rings and Algebras Representation Theory 16E45 |
| url | https://arxiv.org/abs/2503.22493 |