2023/06/01 by Alain Connes, Connes, Alain, Caterina Consani +1
Computer Science · Mathematics · #11R56 #13F35 #14C40 #14G40 #14H05 #18N60 #19D55 #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2306.00456
openalex publication_date 2023/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that by working over the absolute base \mathbb S (the categorical version of the sphere spectrum) instead of \mathbb S[± 1] improves our previous Riemann-Roch formula for \rm 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+X2.