Riemann-Roch for the ring $\mathbb{Z}$

Abstract

We show that by working over the absolute base $\mathbb{S}$ (the categorical version of the sphere spectrum) instead of $\mathbb{S}[\pm 1]$ improves our previous Riemann-Roch formula for $\overline{\mathrm{Spec}\mathbb{Z}}$. The formula equates the (integer-valued) Euler characteristic of an Arakelov divisor with the sum of the degree of the divisor (using logarithms with base 2) and the number $1$, thus confirming the understanding of the ring $\mathbb{Z}$ as a ring of polynomials in one variable over the absolute base $\mathbb{S}$, namely $\mathbb{S}[X],1+1=X+X^2$.

9 pages, 1 Figure