# memo_has_moved_text();New foundations for geometry: Two non-additive languages for arithmetical geometry

M. J. Shai Haran

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1166
ISBNs: 978-1-4704-2312-4 (print); 978-1-4704-3641-4 (online)
DOI: https://doi.org/10.1090/memo/1166
Published electronically: December 6, 2016

View full volume PDF

View other years and numbers:

Chapters

• Introduction

Part 1. $\mathbb F$-$\mathcal R$ings

• Chapter 1. Definition of $\mathbb F$-$\mathcal R$ings
• Appendix A.
• Appendix B. Examples of $\mathbb F$-$\mathcal R$ings
• Appendix A. Proof of Ostrowski’s theorem
• Appendix B. Geometry
• Appendix C. Symmetric Geometry
• Appendix D. Pro - limits
• Appendix E. Vector bundles
• Appendix F. Modules

Part 2. Generalized Rings

• Appendix G. Generalized Rings
• Appendix H. Ideals
• Appendix I. Primes and Spectra
• Appendix J. Localization and sheaves
• Appendix K. Schemes
• Appendix L. Products
• Appendix M. Modules and differentials
• Appendix A. Beta integrals and the local factors of zeta

### Abstract

We give two simple generalizations of commutative rings. They form (co)-complete categories, that contain commutative (semi-) rings (e.g. with the usual multiplication ). But they also contains the "integers" (and ), and the "residue fields" (and ), of the real (and complex) numbers. Here is the collection of unit balls, and is the collection of spheres augmented with a . The initial object is "the field with one element" .One generalization, - the "commutative generalized rings", is an axiomatization of finitely generated free modules over a commutative ring, together with the operations of multiplication and contraction. This is the more geometric language: for any we associate its (symmetric) spectrum, , a compact Zariski space, with a sheaf of over it. By glueing such spectra we get generalized schemes , a full sub-category of the locally-generalized-ringed-spaces. For a number field , with the ring of integers , the compatification of is a pro-object , and its points are the valuation-sub- of : .For , we have a (co)-complete abelian category of - modules with enough injectives and projectives. For in , we obtain the - module of Kähler differentials , satisfying all the usual properties. We compute the universal derivation .All these remain true for the second generalization - the "commutative with involution", the axiomatization of the category of finitely generated free -modules with -linear maps, and the operations of composition,direct sum, and taking transpose.This is the more "linear", or K-theoretic language: for , we have its algebraic K-theory spectum: , and for we obtain the sphere spectrum .For a compact valuation we associate a "zeta" function, so that we obtain the usual factor for the p-adic integers , while we get for the real integers .For , we define the category of vector bundles over , by a certain completion of the categories of vector bundles on the finite layers . For a number field , the isomorphism classes of rank vector bundles over are in natural bijection withwhere (resp. ) for real (resp. complex) place of . E.g. for : , and for : .We have the following "commutative" diagram of adjunctions:where is the left adjoint of the forgetfull functor and .We describe the ordinary commutative (semi)- ring associated by the right adjoint functor to the - fold tensor product (resp. ).Its elements are (non-uniquely) represented as , where are finite rooted trees, with maps , and is a bijection of their leaves , and for we have in addition signs .

### References [Enhancements On Off] (What's this?)

• [A] S. Ju. Arakelov, An intersection theory for divisors on an arithmetic surface, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1179–1192 (Russian). MR 0472815
• [AM] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
• [BG] Kenneth S. Brown and Stephen M. Gersten, Algebraic $K$-theory as generalized sheaf cohomology, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 266–292. Lecture Notes in Math., Vol. 341. MR 0347943
• [Bo09] J. Borger Lambda- rings and the field with one element, arxiv: 0906.3146 (2009)
• [CC09] Allain Connes, Caterina Consani : Schemes over and zeta functions, arXiv:0903.2024v3
• [CC14] Allain Connes, Caterina Consani : The Arithmetical site, arXiv:1405.4527v
• [CCM] Alain Connes, Caterina Consani, Matilde Marcolli : Fun with , arXiv:0806.2401
• [CF] J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. Reprint of the 1967 original. MR 911121
• [De] Anton Deitmar, Schemes over $\Bbb F_1$, Number fields and function fields—two parallel worlds, Progr. Math., vol. 239, Birkhäuser Boston, Boston, MA, 2005, pp. 87–100. MR 2176588, https://doi.org/10.1007/0-8176-4447-4_6
• [Du] N. Durov: New approach to Arakelov geometry arxiv: 0704.2030
• [EGA] A. Grothendieck, J. Dieudonné: Eléments de Géométrie Algébrique, Publ. Math. IHES 4 (1960); 8 (1961); 11 (1961); 17 (1963); 20 (1964); 24 (1965); 28 (1966); 32 (1967).
• [F91] Gerd Faltings, Lectures on the arithmetic Riemann-Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton University Press, Princeton, NJ, 1992. Notes taken by Shouwu Zhang. MR 1158661
• [H89] Shai Haran, Index theory, potential theory, and the Riemann hypothesis, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 257–270. MR 1110396, https://doi.org/10.1017/CBO9780511526053.010
• [H01] M. J. Shai Haran, The mysteries of the real prime, London Mathematical Society Monographs. New Series, vol. 25, The Clarendon Press, Oxford University Press, New York, 2001. MR 1872029
• [H07] M. J. Shai Haran, Non-additive geometry, Compos. Math. 143 (2007), no. 3, 618–688. MR 2330442, https://doi.org/10.1112/S0010437X06002624
• [H08] Shai M. J. Haran, Arithmetical investigations, Lecture Notes in Mathematics, vol. 1941, Springer-Verlag, Berlin, 2008. Representation theory, orthogonal polynomials, and quantum interpolations. MR 2433635
• [H09] S. Haran: Prolegomena to any future Arithmetic that will be able to present itself as a Geometry, arXiv: mathAG/0911.3522.
• [H10] Shai M. J. Haran, Invitation to nonadditive arithmetical geometry, Casimir force, Casimir operators and the Riemann hypothesis, Walter de Gruyter, Berlin, 2010, pp. 249–265. MR 2777720
• [Hart] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
• [Ho] Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
• [I] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
Luc Illusie, Complexe cotangent et déformations. II, Lecture Notes in Mathematics, Vol. 283, Springer-Verlag, Berlin-New York, 1972 (French). MR 0491681
• [KOW] Nobushige Kurokawa, Hiroyuki Ochiai, and Masato Wakayama, Absolute derivations and zeta functions, Doc. Math. Extra Vol. (2003), 565–584. Kazuya Kato’s fiftieth birthday. MR 2046608
• [Lo12] Oliver Lorscheid, Blueprints—towards absolute arithmetic?, J. Number Theory 144 (2014), 408–421. MR 3239169, https://doi.org/10.1016/j.jnt.2014.04.006
• [M] Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 4, 121–163. Columbia University Number Theory Seminar (New York, 1992). MR 1330931
• [Mac] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
• [O] Uri Onn, From $p$-adic to real Grassmannians via the quantum, Adv. Math. 204 (2006), no. 1, 152–175. MR 2233130, https://doi.org/10.1016/j.aim.2005.05.012
• [PL] J.L. Pena, O. Lorscheid: Mapping -land: An overview of geometries over the field with one element, preprint, arXiv: mathAG/0909.0069.
• [Q67] Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
• [Q69] Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 0258031, https://doi.org/10.2307/1970725
• [Q70] Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
• [Q73] Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129
• [S] Christophe Soulé, Les variétés sur le corps à un élément, Mosc. Math. J. 4 (2004), no. 1, 217–244, 312 (French, with English and Russian summaries). MR 2074990
• [SABK] C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. MR 1208731
• [Seg] Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 0353298, https://doi.org/10.1016/0040-9383(74)90022-6
• [Sm] A. L. Smirnov, Hurwitz inequalities for number fields, Algebra i Analiz 4 (1992), no. 2, 186–209 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 2, 357–375. MR 1182400
• [Tak12] S. Takagi: Compactifying , preprint, arXiv: mathAG/1203.4914.
• [TV] B. Toën, M. Vaquié: Au-dessous de , preprint, arXiv: math/0509684.
• [W39] A. Weil: Sur l'analogie entre les corps de nombres algébriques et les corps de fonctions algébriques [], Oevres Scient. I, 236-240, (1980), Springer-Verlag; Revue Scientifique 77 (1939) 104-106.