Deitmar's versus Toen-Vaquie's schemes over F_1

TL;DR

This paper proves the equivalence of two different definitions of schemes over the field with one element, F_1, and derives new results in monoid algebra, bridging different perspectives in algebraic geometry.

Contribution

It establishes the equivalence between Deitmar's and Toen-Vaquie's schemes over F_1, providing a unified framework and new foundational results in monoid algebra.

Findings

Proved the equivalence of two scheme definitions over F_1

Derived new basic results in commutative monoid algebra

Established a symmetry with classical scheme theory

Abstract

We show the equivalence between Deitmar's and Toen-Vaquie's notions of schemes over F_1 (the 'field with one element'), establishing a symmetry with the classical case of schemes, seen either as spaces with a structure sheaf, or functors of points. In proving so, we also conclude some new basic results on commutative algebra of monoids.