The combinatorial-motivic nature of $\mathbb{F}_1$-schemes

TL;DR

This paper explores the combinatorial and motivic aspects of $un$-schemes, reviewing foundational theories, introducing a new framework called $$-schemes, and connecting these ideas to motives, zeta functions, and hyperring extensions.

Contribution

It introduces the author's version of $un$-schemes ($$-schemes), extending the combinatorial study and linking it to motives, zeta functions, and hyperring theory.

Findings

Combinatorial study of $un$-schemes via generalized graph theory.

Introduction of $$-schemes as a new framework.

Connections between hyperring extensions and group actions on projective spaces.

Abstract

We review Deitmar's theory of monoidal schemes to start with, and have a detailed look at the standard examples. It is explained how one can combinatorially study such schemes through a generalization of graph theory. In a more general setting we then introduce the author's version of $\mathbb{F}_1$-schemes (called $\Upsilon$-schemes here), after which we study Grothendieck's motives in some detail in order to pass to "absolute motives". Throughout several considerations about absolute zeta functions are written. In a final part of the chapter, we describe the approach of Connes and Consani to understand the ad`ele class space through hyperring extension theory, in which a marvelous connection is revealed with certain group actions on projective spaces, and brand new results in the latter context are described. Many questions are posed, conjectures are stated and speculations are made.