Topological and scheme-theoretic properties of the $D$-graded Proj construction

TL;DR

This paper extends the topological and scheme-theoretic understanding of the Proj construction from the classical single grading to multigraded rings, introducing new concepts like D-prime ideals and a multigraded Nullstellensatz.

Contribution

It generalizes the Proj construction to multigraded rings by characterizing D-primeness and establishing a multigraded Nullstellensatz, along with criteria for separability and Serre twist freeness.

Findings

Characterization of multigraded Proj using D-prime ideals

Establishment of a multigraded Nullstellensatz

Conditions for separability and Serre twist freeness

Abstract

We generalize the topological description of the $\mathbb{N}$-graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups $D$. However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using $D$-prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj$^D(S)$ is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their freeness.