On Multi-Graded Proj Schemes

Abstract

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.