发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
在康恩斯 - 康萨尼框架下，利用源自同伦论的工具，将\(\mathbb{F}_1\) - 代数作为\(\Gamma\) - 集范畴中的幺半群对象，发展交换代数和代数几何基本概念，给出\(\mathbb{F}_1\) - 代数局部化理论等。
AI 中文摘要
我们在康恩斯 - 康萨尼框架下发展了\(\mathbb{F}_1\)上交换代数和代数几何的基本概念，将\(\mathbb{F}_1\) - 代数建模为\(\Gamma\) - 集范畴中的幺半群对象。我们的主要贡献是\(\mathbb{F}_1\) - 代数的局部化理论以及交换\(\mathbb{F}_1\) - 代数\(A\)的素谱\(\Spec A\)的构造。然后证明对于任何绝对仿射概型\(X = \Spec A\)，\(\Gamma(X, \mathcal{O}_X) = A\)，并建立交换\(\mathbb{F}_1\) - 代数范畴与绝对仿射概型范畴之间的反等价关系。
英文摘要
We construct an affine geometry for commutative $\mathbb F_1$-algebras in the framework of Connes and Consani. Using their localization construction, we equip the Deitmar spectrum of the underlying multiplicative monoid $A(1_+)$ with a sheaf $\mathcal O_A$ of $\mathbb F_1$-algebras. We prove that $\mathcal O_A(D(f))\cong A_f$ and, in particular, that global sections recover $A$. The resulting spectrum construction yields an anti-equivalence between commutative $\mathbb F_1$-algebras and absolute affine schemes. We compare our choice of covering families with the Connes--Consani topology and explain its role in affine reconstruction. Finally, we describe base change to $\mathbb Z$, under which the absolute affine scheme associated to the Eilenberg--MacLane algebra $HR$ recovers the classical affine scheme $\operatorname{Spec}R$.
Comments25 pages
