New building blocks for $\mathbb{F}_1$-geometry: bands and band schemes

TL;DR

The paper introduces bands as a ring-like generalization bridging rings and idylls, and develops both algebraic and geometric theories: a basic commutative-algebra framework for bands (ideals, localizations, quotients, limits) and a geometric theory of band schemes built from the spectrum of bands. It then constructs band schemes and analyzes their functorial relations to ordinary schemes, monoid schemes, and ordered blue schemes, including base extension and fiber products. A key innovation is the visualization toolkit for band schemes, featuring the kernel space, null space, and Tits space, which connect to matroid theory, toric and tropical geometry, and Weyl groups; these visualizations provide a conceptual bridge to ${\mathbb F}_1$-geometry and broad applications such as toric models and tropicalization. Together, the results establish a robust framework for algebraic geometry over bands, with concrete instances and links to matroid theory, toric geometry, and group-theoretic structures, highlighting the potential for ${\mathbb F}_1$-geometry and tropicalization paradigms.

Abstract

We develop and study a generalization of commutative rings called bands, along with the corresponding geometric theory of band schemes. Bands generalize both hyperrings, in the sense of Krasner, and partial fields in the sense of Semple and Whittle. They from a ring-like counterpart to the field-like category of idylls introduced by the first and third author. The first part of the paper is dedicated to establishing fundamental properties of bands analogous to basic facts in commutative algebra. In particular, we introduce various kinds of ideals in a band and explore their properties, and we study localization, quotients, limits, and colimits. The second part of the paper studies band schemes. After giving the definition, we present some examples of band schemes, along with basic properties of band schemes and morphisms thereof, and we describe functors into some other scheme theories. In the third part, we discuss some ``visualizations'' of band schemes, which are different topological spaces that one can functorially associate to a band scheme $X$.

Figures (2)

• Figure 1: A commutative diagram of fully faithful embeddings of categories
• Figure 2: Embedding of the Coxeter complex ${\mathcal{C}}_3$ into $\Omega_{S_3}$.

Theorems & Definitions (143)

• Example
• Definition 1
• Lemma 1
• proof
• Definition 2
• Lemma 2
• proof
• Definition 3
• Lemma 3
• proof