On multi-graded Proj schemes

Arnaud Mayeux; Simon Riche

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arXiv · 2310.13502

On multi-graded Proj schemes

Arnaud Mayeux·Simon Riche

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Abstract

We review the construction (due to Brenner--Schr\"oer) 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--Schr\"oer. 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.

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Arnaud Mayeux, Simon Riche. 2023-10-20. On multi-graded Proj schemes. https://arxiv.org/abs/2310.13502

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