On multi-graded Proj schemes
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910874393903104 |
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| author | Mayeux, Arnaud Riche, Simon |
| author_facet | Mayeux, Arnaud Riche, Simon |
| contents | We review the construction (due to Brenner--Schröer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for $\mathbb{N}$-graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the $\mathbb{N}$-graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner--Schröer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13502 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On multi-graded Proj schemes Mayeux, Arnaud Riche, Simon Algebraic Geometry We review the construction (due to Brenner--Schröer) of the Proj scheme associated with a ring graded by a finitely generated abelian group. This construction generalizes the well-known Grothendieck Proj construction for $\mathbb{N}$-graded rings; we extend some classical results (in particular, regarding quasi-coherent sheaves on such schemes) from the $\mathbb{N}$-graded setting to this general setting, and prove new results that make sense only in the general setting of Brenner--Schröer. Finally, we show that flag varieties of reductive groups, as well as some vector bundles over such varieties attached to representations of a Borel subgroup, can be naturally interpreted in this formalism. |
| title | On multi-graded Proj schemes |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2310.13502 |