Derived equivalences, new matrix equivalences, and homological conjectures
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911260693495808 |
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| author | Li, Xiaogang Xi, Changchang |
| author_facet | Li, Xiaogang Xi, Changchang |
| contents | Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of \textbf{two} matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost $ν$-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for $p$-regular parts and for $p$-singular parts. As an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26353 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derived equivalences, new matrix equivalences, and homological conjectures Li, Xiaogang Xi, Changchang Representation Theory Rings and Algebras Based on the fact that every finite-dimensional algebra over a field is isomorphic to the centralizer of \textbf{two} matrices, we approach the representation theory of finite-dimensional algebras over fields by centralizers of matrices. The first fundamental question is to study the centralizer of a single matrix, called a centralizer matrix algebra. By introducing three new equivalence relations on all square matrices over a field, we completely characterize Morita, derived and almost $ν$-stable derived equivalences between centralizer matrix algebras in terms of these matrix equivalences, respectively. Further, we show that a derived equivalence between centralizer matrix algebras of permutation matrices induces both a Morita equivalence and additional derived equivalences for $p$-regular parts and for $p$-singular parts. As an application, we show that the finitistic dimension conjecture and the Nakayama conjecture are valid for centralizer matrix algebras. |
| title | Derived equivalences, new matrix equivalences, and homological conjectures |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2509.26353 |