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

Alain Connes; Caterina Consani

doi:10.5802/crmath.543

Research article - Number theory

Riemann–Roch for the ring

ℤ

Alain Connes ^1,² ; Caterina Consani ³

¹ Collège de France, 3 rue d’Ulm F-75005 Paris, France
² IHES, 35 Rte de Chartres, 91440 Bures-sur-Yvette, France
³ Department of Mathematics, Johns Hopkins University, Baltimore MD 21218, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 229-235

Abstract (Original language)
Résumé

We show that by working over the absolute base $𝕊$ (the categorical version of the sphere spectrum) instead of $𝕊 [\pm 1]$ improves our previous Riemann–Roch formula for $\bar{Spec ℤ}$ . 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 $ℤ$ as a ring of polynomials in one variable over the absolute base $𝕊$ , namely $𝕊 [X], 1 + 1 = X + X^{2}$ .

Nous montrons que l’utilisation de la base absolue $𝕊$ (la version catégorique du spectre en sphère) au lieu de $𝕊 [\pm 1]$ , améliore notre formule de Riemann–Roch précédente pour $\bar{Spec ℤ}$ . La formule calcule la caractéristique d’Euler (à valeur entière) d’un diviseur d’Arakelov comme la somme du degré du diviseur (en utilisant des logarithmes de base 2) et le nombre $1$ , confirmant ainsi la compréhension de l’anneau $ℤ$ comme un anneau de polynômes en une variable sur la base absolue $𝕊$ ,à savoir $𝕊 [X], 1 + 1 = X + X^{2}$ .

Received: 2023-06-05
Revised: 2023-07-06
Accepted: 2023-07-21
Published online: 2024-05-02

DOI: 10.5802/crmath.543

Classification: 14C40, 14G40, 14H05, 11R56, 13F35, 18N60, 19D55

Author's affiliations:

Alain Connes ^{1, 2}; Caterina Consani ³

¹ Collège de France, 3 rue d’Ulm F-75005 Paris, France
² IHES, 35 Rte de Chartres, 91440 Bures-sur-Yvette, France
³ Department of Mathematics, Johns Hopkins University, Baltimore MD 21218, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{CRMATH_2024__362_G3_229_0,
     author = {Alain Connes and Caterina Consani},
     title = {Riemann{\textendash}Roch for the ring $\mathbb{Z}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {229--235},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     doi = {10.5802/crmath.543},
     language = {en},
}

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Alain Connes; Caterina Consani. Riemann–Roch for the ring $\mathbb{Z}$. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 229-235. doi: 10.5802/crmath.543

References
Cited by

[1] A. Connes; C. Consani Absolute algebra and Segal’s $γ$ -rings: au dessous de $\bar{Spec (ℤ)}$ , J. Number Theory, Volume 162 (2016), pp. 518-551 | DOI | Zbl | MR

[2] A. Connes; C. Consani Riemann-Roch for $\bar{Spec ℤ}$ (2023) (to appear in Bulletin des Sciences Mathématiques) | arXiv

[3] Bjørn Ian Dundas; Thomas G. Goodwillie; Randy McCarthy The local structure of algebraic K-theory, Algebra and Applications, 18, Springer, 2013 | DOI | Zbl

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