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

TL;DR

This paper introduces bands, a new algebraic structure generalizing hyperrings and partial fields, and develops their geometric theory through band schemes, expanding the landscape of algebraic geometry beyond classical rings.

Contribution

It defines and studies bands and band schemes, establishing foundational properties and exploring their geometric and topological aspects, thus broadening algebraic geometry frameworks.

Findings

Bands generalize hyperrings and partial fields.

Fundamental properties of bands analogous to commutative algebra.

Examples and basic properties of band schemes.

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 $X$.