On multi-graded Proj schemes

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.