Non-Additive Geometry and Frobenius Correspondences

TL;DR

This paper introduces new algebraic frameworks using vectors, bi-operads, and matrices to address the limitations of traditional algebraic geometry in arithmetical contexts, especially at the real prime.

Contribution

It develops two novel languages for arithmetical geometry that replace classical ring structures with vector and matrix-based systems, simplifying the handling of Frobenius correspondences.

Findings

New algebraic languages based on vectors and matrices

Matrices up to conjugation yield new commutative rings with Frobenius endomorphisms

Simplification of algebraic geometry at the real prime

Abstract

The usual language of algebraic geometry is not appropriate for Arithmetical geometry: addition is singular at the real prime. We developed two languages that overcome this problem: one replace rings by the collection of "vectors" or by bi-operads and another based on "matrices" or props. These are the two languages of [Har17], but we omit the involutions which brings considerable simplifications. Once one understands the delicate commutativity condition one can proceed following Grothendieck footsteps exactly. The square matrices, when viewed up to conjugation, give us new commutative rings with Frobenius endomorphisms.