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

25 pages; inclusion of corollary 4.6, new abstract, minor corrections