Computing Direct Sum Decompositions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866917539072704512 |
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| author | Mallory, Devlin Sayrafi, Mahrud |
| author_facet | Mallory, Devlin Sayrafi, Mahrud |
| contents | We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when R is local and k is a finite field, opening the door to computing decompositions in singularity theory. We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards (including in the non-graded case) and syzygies over Artinian rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19799 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing Direct Sum Decompositions Mallory, Devlin Sayrafi, Mahrud Commutative Algebra Algebraic Geometry 16D70, 14F06, 13A35 We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when R is local and k is a finite field, opening the door to computing decompositions in singularity theory. We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards (including in the non-graded case) and syzygies over Artinian rings. |
| title | Computing Direct Sum Decompositions |
| topic | Commutative Algebra Algebraic Geometry 16D70, 14F06, 13A35 |
| url | https://arxiv.org/abs/2412.19799 |