Some computational aspects of spectral sequences in Čech cohomology
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916775296237568 |
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| author | Zach, Matthias |
| author_facet | Zach, Matthias |
| contents | Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily challenges the capacities of modern computers. We describe an algorithm and its implementation to compute a spectral sequence converging to the higher direct images of a bounded complex of sheaves on a product of projective spaces $\mathbb P = \mathbb P^{r_1}\times \dots \times \mathbb P^{r_m}$ over an arbitrary affine base $\mathrm{Spec} R$. We assume the ring $R$ to be computable and the complex of sheaves to be represented by an actual complex of (multi-)graded modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some computational aspects of spectral sequences in Čech cohomology Zach, Matthias Algebraic Geometry Commutative Algebra 14Q20, 14-04, 13P20, 13D45, 68W30 Sheaf cohomology or, more generally, higher direct images of coherent sheaves along proper morphisms are central to modern algebraic geometry. However, the computation of these objects is a non-trivial and expensive task which easily challenges the capacities of modern computers. We describe an algorithm and its implementation to compute a spectral sequence converging to the higher direct images of a bounded complex of sheaves on a product of projective spaces $\mathbb P = \mathbb P^{r_1}\times \dots \times \mathbb P^{r_m}$ over an arbitrary affine base $\mathrm{Spec} R$. We assume the ring $R$ to be computable and the complex of sheaves to be represented by an actual complex of (multi-)graded modules. |
| title | Some computational aspects of spectral sequences in Čech cohomology |
| topic | Algebraic Geometry Commutative Algebra 14Q20, 14-04, 13P20, 13D45, 68W30 |
| url | https://arxiv.org/abs/2506.02636 |