Riemann–Roch for the ring $\mathbb{Z}$
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 ${\overline{\operatorname{Spec}\mathbb{Z}}}$. The formula equates the (integer valued) Euler characteristic of...
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| Language: | English |
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.543/ |
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| author | Connes, Alain Consani, Caterina |
| author_facet | Connes, Alain Consani, Caterina |
| author_sort | Connes, Alain |
| collection | DOAJ |
| description | 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 ${\overline{\operatorname{Spec}\mathbb{Z}}}$. 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 $\mathbb{S}[X], 1+1=X+X^2$. |
| format | Article |
| id | doaj-art-825459f6c6184603890323a8c17914d7 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-825459f6c6184603890323a8c17914d72025-02-07T11:19:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G322923510.5802/crmath.54310.5802/crmath.543Riemann–Roch for the ring $\mathbb{Z}$Connes, Alain0Consani, Caterina1Collège de France, 3 rue d’Ulm F-75005 Paris, France; IHES, 35 Rte de Chartres, 91440 Bures-sur-Yvette, FranceDepartment of Mathematics, Johns Hopkins University, Baltimore MD 21218, USAWe 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 ${\overline{\operatorname{Spec}\mathbb{Z}}}$. 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 $\mathbb{S}[X], 1+1=X+X^2$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.543/ |
| spellingShingle | Connes, Alain Consani, Caterina Riemann–Roch for the ring $\mathbb{Z}$ Comptes Rendus. Mathématique |
| title | Riemann–Roch for the ring $\mathbb{Z}$ |
| title_full | Riemann–Roch for the ring $\mathbb{Z}$ |
| title_fullStr | Riemann–Roch for the ring $\mathbb{Z}$ |
| title_full_unstemmed | Riemann–Roch for the ring $\mathbb{Z}$ |
| title_short | Riemann–Roch for the ring $\mathbb{Z}$ |
| title_sort | riemann roch for the ring mathbb z |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.543/ |
| work_keys_str_mv | AT connesalain riemannrochfortheringmathbbz AT consanicaterina riemannrochfortheringmathbbz |