Torsor and Quotient Presentations for $D$-homogeneous Spectra
Felix Göbler
TL;DR
The paper generalizes the Proj construction to multigraded rings by defining $\mathrm{Proj}^D(S)$ for a finitely generated abelian group $D$ and the notion of relevant elements, enabling a coordinate description from degree-zero localizations. It shows that each patch $\mathrm{Spec}(S_{(f)})$ with $f$ relevant is a geometric quotient of $\mathrm{Spec}(S_f)$ by the diagonalizable group $G=\mathrm{Spec}(S_0[D])$, and that the global object satisfies $\mathrm{Proj}^D(S)\cong D(S_+)\sslash G$. The paper provides necessary and sufficient conditions for $\pi_f$ to be a pseudo $G$-torsor (and locally trivial after base change) in terms of the unit group degrees and integrality, and introduces strong relevance via $D^f=D$ with analysis of the torsion-free part of $D$. These results extend Cox-type quotient descriptions from toric geometry to general multigraded rings, connecting to toric varieties and Mori dream spaces, and establishing a quotient-centered foundation for $D$-graded projective geometry.
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 $π_f: {\rm Spec}(S_f) \to {\rm Spec}(S_{(f)})$ 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.