Luqiao Xu | Searcharxiv

arXiv2026

Localization and Affine Schemes over $\mathbb{F}_1$

We develop the basic notions of commutative algebra and algebraic geometry over the field with one element $\mathbb{F}_1$, working within the Connes-Consani framework, which models $\mathbb{F}_1$-algebras as monoid objects in the category of $Γ$-sets. In this setting, $\mathbb{F}_1$-algebras generalize commutative rings by encoding the algebraic structure functorially, using machinery originating in homotopy theory. Our main contribution is a theory of localization for $\mathbb{F}_1$-algebras and the construction of prime spectrum $\Spec A$ for a commutative $\mathbb{F}_1$-algebra $A$. We then prove that $Γ(X, \mathcal{O}_X) = A$ for any absolute affine scheme $X=\Spec A$ and establish an anti-equivalence between the category of commutative $\mathbb{F}_1$-algebras and the category of absolute affine schemes.

math.AG↗

arXiv2026

Hyper-Operations and Extension of Scalars from $\mathbb{F}_1$ to $\mathbb{Z}$

The additive structure of $\mathbb{F}_1$-modules (in the sense of Segal's $Γ$-sets) differs fundamentally from that of abelian groups: addition is encoded through a family of $n$-ary hyper-operations that are multivalued and do not satisfy classical associativity. We establish a \emph{law of generalized associativity} showing that, despite this failure of strict associativity, all $n$-ary sums are controlled by successive binary operations. This enables us to construct an extension of scalars functor $-\otimes_{\mathbb{F}_1} \mathbb{Z}: \mathbb{F}_1\mathbf{Mod} \to \mathbf{Ab}$ that universally strictifies the hyper-additive structure of $\mathbb{F}_1$-modules into classical abelian group addition. We prove this functor is left adjoint to the Eilenberg-MacLane functor $H: \mathbf{Ab} \to \mathbb{F}_1\mathbf{Mod}$. Extending to the multiplicative setting, we obtain an adjunction $-\otimes_{\mathbb{F}_1} \mathbb{Z}: \mathbb{F}_1\mathbf{Alg} \leftrightarrows \mathbf{CRing} : H$ between commutative $\mathbb{F}_1$-algebras and commutative rings. This recovers Deitmar's monoid ring construction for spherical monoid algebras and provides a base change mechanism needed for absolute algebraic geometry.

math.AG↗