Torsor and Quotient Presentations for $D$-homogeneous Spectra

TL;DR

This paper extends the classical Proj construction to $D$-graded rings, exploring properties like prime ideals and quotients, and generalizes toric variety constructions via torsor and quotient presentations.

Contribution

It introduces a comprehensive framework for $D$-graded Proj constructions, including prime ideal distinctions, quotient descriptions, and conditions for geometric quotients and torsors.

Findings

Characterization of $D$-prime ideals in $D$-graded rings

Description of quotients by associated group schemes

Conditions for geometric quotients and torsors

Abstract

The $D$-graded Proj construction provides a general framework for constructing schemes from rings graded by finitely generated abelian groups $D$, yet its properties and applications remain underdeveloped compared to the classical $\mathbb{N}$-graded case. This paper establishes the essential characteristics of $D$-graded rings $S$, like the distinction between $D$-homogeneous prime ideals and $D$-prime ideals if $D$ has torsion. We particularly focus on describing the quotient by the associated group scheme, generalizing the construction of a toric variety from its Cox ring. As in the $\mathbb{N}$-graded construction, the basic affine opens of the Proj construction are given in terms of degree-zero localizations $S_{(f)}$, where $f$ in $S$ homogeneous is \emph{relevant}. We prove that $\pi_f: {\rm Spec}(S_f) \to {\rm Spec}(S_{(f)})$ is a geometric quotient under mild finiteness assumptions if $f$ is relevant, and give necessary and sufficient conditions for this map to be a pseudo ${\rm Spec}(S_0[D])$-torsor.