Strong congruence spaces and dimension in $\mathbb{F}_1$-geometry

TL;DR

This paper introduces strong congruence spaces as a new way to define and analyze dimension in monoid schemes, linking algebraic and geometric properties in $\\mathbb{F}_1$-geometry.

Contribution

It defines strong congruence spaces and demonstrates their equivalence in dimension with complex toric varieties for toric monoid schemes.

Findings

Strong congruence spaces provide a meaningful notion of dimension.

For toric monoid schemes, their strong congruence space matches the dimension of associated complex toric varieties.

The study bridges monoid schemes and classical algebraic geometry.

Abstract

We introduce strong congruence spaces, which are topological spaces that provide a useful concept of dimension for monoid schemes. We study their properties and show that, given a toric monoid scheme over an algebraically closed basis, its strong congruence space and the complex toric variety associated to its fan have the same dimension.