Relevant maps and the algebraic skeleton of simplicial toric prevarieties

TL;DR

This paper develops a systematic framework for rational maps between multigraded schemes, linking algebraic structures to geometric objects like toric prevarieties, and introduces relevant subsets to control morphism conditions.

Contribution

It introduces a new approach to rational maps between multigraded rings using relevant subsets, enabling a categorical equivalence with simplicial toric prevarieties.

Findings

Categorical anti-equivalence between triples (D, S, B) and simplicial toric prevarieties.

New notion of relevant subsets B to control morphism conditions.

Multigraded polynomial rings encode combinatorial data for toric geometry.

Abstract

Morphisms between schemes arising from multigraded rings are essential for understanding geometric relationships in algebraic geometry, yet a systematic theory for such maps has been lacking. In this paper, we develop a comprehensive framework for rational maps between multigraded Proj schemes by introducing several notions of maps between their underlying multigraded rings. A key challenge is that to induce actual morphisms (rather than just rational maps), the ring homomorphism $\varphi\colon R \to S$ must hit every relevant element in $S$. To address this, we introduce the use of relevant subsets $B \subseteq S_+$ (where $S_+$ is the ideal generated by all relevant elements), $B \unlhd S$, which allow us to control this condition more flexibly. As an application, we show that multigraded noetherian polynomial rings naturally encode combinatorial data, giving rise to systems of fans and thus to toric prevarieties. By leveraging our notion of rational maps with those relevant subsets, we prove that the category of triples $(D, S, B)$ - where $D$ is a finitely generated abelian group, $S$ is a $D$-graded noetherian polynomial ring, and $B \unlhd S$ is a subset of $S_+$ - together with rational maps of conical rings, is anti-equivalent to the category of simplicial toric prevarieties.