Uniqueness of indecomposable idempotents in algebras with involution
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Karemaker, Valentijn Tamagawa, Akio Yu, Chia-Fu |
| author_facet | Karemaker, Valentijn Tamagawa, Akio Yu, Chia-Fu |
| contents | We prove uniqueness of a decomposition of $1$ into indecomposable Hermitian idempotents in an order of a finite-dimensional $\mathbb{Q}$-algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre and Serre with different methods and over algebraically closed fields. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09323 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of indecomposable idempotents in algebras with involution Karemaker, Valentijn Tamagawa, Akio Yu, Chia-Fu Number Theory Algebraic Geometry 16H20 (14K12 11E39 11G10) We prove uniqueness of a decomposition of $1$ into indecomposable Hermitian idempotents in an order of a finite-dimensional $\mathbb{Q}$-algebra with positive involution, by generalising a result of Eichler on unique decomposition of lattices. We use this result to prove that polarised abelian varieties over any field admit a unique decomposition into indecomposable polarised abelian subvarieties, a result previously shown by Debarre and Serre with different methods and over algebraically closed fields. We prove that an analogous uniqueness result holds true for arbitrary polarised integral Hodge structures, and derive a consequence for their automorphism groups. |
| title | Uniqueness of indecomposable idempotents in algebras with involution |
| topic | Number Theory Algebraic Geometry 16H20 (14K12 11E39 11G10) |
| url | https://arxiv.org/abs/2402.09323 |