Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2024

On multi-graded Proj schemes

Arnaud Mayeux
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Résumé

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.

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Dates et versions

hal-04669439 , version 1 (08-08-2024)
hal-04669439 , version 2 (13-03-2025)

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Arnaud Mayeux, Simon Riche. On multi-graded Proj schemes. 2025. ⟨hal-04669439v2⟩
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