On the metaphysics of $\mathbb F_1$

Alain Connes; Caterina Consani

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arXiv · 2307.06748

On the metaphysics of $\mathbb F_1$

Alain Connes·Caterina Consani

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Abstract

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of $\mathbb S[\mu_{n,+}]$-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring $\mathbb Z$ of the integers that we obtained recently uses the understanding of $\mathbb Z$ as a ring of polynomials $\mathbb S[X]$ in one variable over the absolute base $\mathbb S$, where $1+1=X+X^2$. The absolute base $\mathbb S$ (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious $\mathbb F_1$.

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Alain Connes, Caterina Consani. 2023-07-13. On the metaphysics of $\mathbb F_1$. https://arxiv.org/abs/2307.06748

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