Congruence schemes

TL;DR

This paper introduces congruence schemes, a new algebro-geometric framework that generalizes existing theories like monoid schemes and classical algebraic schemes, and connects to Berkovich's approach and number theory.

Contribution

It defines a new category of algebraic objects called congruence schemes, unifying various existing theories and providing new tools for number theory and non-Archimedean geometry.

Findings

Generalizes F1-theories including monoid and classical schemes

Provides a description of Berkovich subdomains within the new framework

Encodes number-theoretic information through congruence schemes

Abstract

A new category of algebro-geometric objects is defined. This construction is a vast generalization of existing F1-theories, as it contains the the theory of monoid schemes on the one hand and classical algebraic theory, e.g. Grothendieck schemes, on the the other. It also gives a handy description of Berkovich subdomains and thus contains Berkovich's approach to abstract skeletons. Further it complements the theory of monoid schemes in view of number theoretic applications as congruence schemes encode number theoretical information as opposed to combinatorial data which are seen by monoid schemes.