Skip to Main Content


AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution


New foundations for geometry Two non-additive languages for arithmetical geometry

About this Title

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
MSC: Primary 11G99, 11R99, 14A20, 14G40

View full volume PDF

View other years and numbers:

Table of Contents

Chapters

  • Introduction

1. $\mathbb {F}$-$\mathcal {R}$ings

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

A. Generalized Rings

  • G. Generalized Rings
  • H. Ideals
  • I. Primes and Spectra
  • J. Localization and sheaves
  • K. Schemes
  • L. Products
  • M. Modules and differentials
  • 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. $\{0,1\}\subseteq [0,1]\subseteq [0,\infty )$ with the usual multiplication $x+y:=\max \{x,y\}$). But they also contains the "integers" $\mathbb {Z}_{\mathbb {R}}$ (and $\mathbb {Z}_{\mathbb {C}}$), and the "residue fields" $\mathbb {F}_{\mathbb {R}}$ (and $\mathbb {F}_{\mathbb {C}}$), of the real (and complex) numbers. Here $\mathbb {Z}_{\mathbb {R}}$ is the collection of unit $L_2$ balls, and $\mathbb {F}_{\mathbb {R}}$ is the collection of spheres augmented with a $0$. The initial object is "the field with one element" $\mathbb {F}_1$.

One generalization, $\mathcal {GR}_c$ - 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 $A\in \mathcal {GR}_c$ we associate its (symmetric) spectrum, $\operatorname {Spec} A$, a compact Zariski space, with a sheaf of $\mathcal {GR}_c$ over it. By glueing such spectra we get generalized schemes $\mathcal {G}\text {Sch}$, a full sub-category of the locally-generalized-ringed-spaces. For a number field $K$, with the ring of integers $\mathcal {O}_K$, the compatification of $\operatorname {Spec} \mathcal {O}_K$ is a pro-object $\bar {\operatorname {Spec} \mathcal {O}_K}\in \text {pro-}\mathcal {G}\text {Sch}$, and its points are the valuation-sub-$\mathcal {GR}_c$ of $K$: $\operatorname {Val}(K)\equiv \{K; \mathcal {O}_{K,\mathfrak {p}}=S^{-1}_{\mathfrak {p}}\mathcal {O}_K, \; \mathfrak {p}\subseteq \mathcal {O}_K\text { finite prime }; K\cap \sigma ^{-1}\mathbb {Z}_{\mathbb {C}} \;, \sigma :K\hookrightarrow \mathbb {C}\}$.

For $A\in \mathcal {GR}_c$, we have a (co)-complete abelian category of $A$- modules with enough injectives and projectives. For $k\rightarrow A$ in $\mathcal {GR}_c$, we obtain the $A$- module of Kähler differentials $\Omega (A/k)$, satisfying all the usual properties. We compute the universal derivation $d:\mathbb {Z}\rightarrow \Omega (\mathbb {Z}/\mathbb {F}_1)_{[1]}$.

All these remain true for the second generalization $\mathbb {F}\mathcal {R}^t_c$- the "commutative $\mathbb {F}\text {-Rings}$ with involution", the axiomatization of the category of finitely generated free $A$-modules with $A$-linear maps, and the operations of composition,direct sum, and taking transpose.

This is the more "linear", or K-theoretic language: for $A\in \mathbb {F}\mathcal {R}^t_c$, we have its algebraic K-theory spectum: $B(A^{-1}A)\simeq \mathbb {Z}\times BGL_{\infty }(A)^+$, and for $A=\mathbb {F}_1$ we obtain the sphere spectrum $\mathbb {Z}\times B\Sigma _{\infty }^+$.

For a compact valuation $\mathbb {F}\mathcal {R}_c^t$ we associate a "zeta" function, so that we obtain the usual factor $(1-p^{-s})^{-1}$ for the p-adic integers $\mathbb {Z}_p$, while we get $2^{\frac {s}{2}}\Gamma (\frac {s}{2})$ for the real integers $\mathbb {Z}_{\mathbb {R}}$.

For $X=\{X_N\}\in \text {pro-}\mathcal {G}\text {Sch}$, we define the category of vector bundles over $X$, by a certain completion of the categories of vector bundles on the finite layers $X_N$. For a number field $K$, the isomorphism classes of rank $n$ vector bundles over $\bar {\operatorname {Spec}\mathcal {O}_K}$ are in natural bijection with \[ GL_{n}(K)\GL_{n}(\mathbb{A}_{K})/∏_{𝜈}GL_{n}(\mathcal{O}_{K,𝜈}). \] where $GL_n(\mathcal {O}_{K,\nu })=\mathcal {O}(n)$ (resp. $U(n)$) for real (resp. complex) place $\nu$ of $K$. E.g. for $n=1$: $\operatorname {Pic}(\bar {\operatorname {Spec}\mathcal {O}_K})=K^*\backslash \mathbb {A}^*_K/ \prod _{\nu }\mathcal {O}^*_{K,\nu }$, and for $K=\mathbb {Q}$: $\operatorname {Pic}(\bar {\operatorname {Spec} \mathbb {Z}})=\mathbb {R}^+$.

We have the following "commutative" diagram of adjunctions: \[ \xymatrix@C=2.5pc @R=2.5pc&\mathcal{GR}_{c}\ar@/_{/}[d]_{ \mathbb{N}\underset{\mathbb{F}}⊗}\ar@/^{1}pc/@^{(}->[r]^{F}&\mathbb{F}\mathcal{R}^{t}_{c}\ar@/^{/}@->>[l]_{U}\ar@/_{/}[d]_{\mathbb{N}\underset{\mathbb{F}}⊗} C\mathcal{R}ig\ar@/_{/}[d]_{K}\ar@/^{1}pc/@^{(}->[r] &\mathbb{N}\\mathcal{GR}_{c}\ar@^{(}->@/_{/}[u]\ar@/_{/}[d]_{ \mathbb{Z}\underset{\mathbb{N}}⊗}\ar@/^{/}@->>[l]\ar@/^{1}pc/@^{(}->[r]^{F}& \mathbb{N}\\mathbb{F}\mathcal{R}_{c}^{t}\ar@^{(}->@/_{/}[u]\ar@/_{/}[d]_{ \mathbb{Z}\underset{\mathbb{N}}⊗}\ar@/^{/}@->>[l]_{U} \text{CRing}\ar@^{(}->@/_{/}[u]\ar@/^{1}pc/@^{(}->[r]&\mathbb{Z}\\mathcal{GR}_{c}\ar@^{(}->@/_{/}[u]\ar@/^{/}@->>[l]\ar@/^{1}pc/@^{(}->[r]^{F}&\mathbb{Z}\\mathbb{F}\mathcal{R}^{t}_{c}\ar@/^{/}@->>[l]_{U}\ar@^{(}->@/_{/}[u] \] where $F$ is the left adjoint of the forgetfull functor $U$ and $U\circ F=id$.

We describe the ordinary commutative (semi)- ring associated by the right adjoint functor to the $n$- fold tensor product $\mathbb {Z}\underset {\mathbb {F}_{\pm 1}}{\otimes }\dots \underset {\mathbb {F}_{\pm 1}}{\otimes } \mathbb {Z}$ (resp. $\mathbb {N}\underset {\mathbb {F}}{\otimes }\dots \underset {\mathbb {F}}{\otimes }\mathbb {N}$).

Its elements are (non-uniquely) represented as $(F,G,\sigma )$, where $F,G$ are finite rooted trees, with maps $F\setminus \partial F,G\setminus \partial G\rightarrow \{1,2,\dots ,n\}$, and $\sigma$ is a bijection of their leaves $\sigma :\partial F\xrightarrow {\sim }\partial G$, and for $\mathbb {Z}$ we have in addition signs $\epsilon : \partial F\rightarrow \{\pm 1\}$.

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

References
  • 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
  • M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
  • 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
  • J. Borger Lambda- rings and the field with one element, arxiv: 0906.3146 (2009)
  • Allain Connes, Caterina Consani : Schemes over $\mathbb {F}_1$ and zeta functions, arXiv:0903.2024v3
  • Allain Connes, Caterina Consani : The Arithmetical site, arXiv:1405.4527v
  • Alain Connes, Caterina Consani, Matilde Marcolli : Fun with $\mathbb {F}_1$, arXiv:0806.2401
  • 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
  • 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, DOI 10.1007/0-8176-4447-4_{6}
  • N. Durov: New approach to Arakelov geometry arxiv: 0704.2030
  • 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).
  • 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
  • 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, DOI 10.1017/CBO9780511526053.010
  • 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
  • M. J. Shai Haran, Non-additive geometry, Compos. Math. 143 (2007), no. 3, 618–688. MR 2330442, DOI 10.1112/S0010437X06002624
  • 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
  • S. Haran: Prolegomena to any future Arithmetic that will be able to present itself as a Geometry, arXiv: mathAG/0911.3522.
  • 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
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • Mark Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
  • Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
  • 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
  • Oliver Lorscheid, Blueprints—towards absolute arithmetic?, J. Number Theory 144 (2014), 408–421. MR 3239169, DOI 10.1016/j.jnt.2014.04.006
  • 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
  • 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
  • Uri Onn, From $p$-adic to real Grassmannians via the quantum, Adv. Math. 204 (2006), no. 1, 152–175. MR 2233130, DOI 10.1016/j.aim.2005.05.012
  • J.L. Pena, O. Lorscheid: Mapping ${\mathbb F}_1$-land: An overview of geometries over the field with one element, preprint, arXiv: mathAG/0909.0069.
  • Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
  • Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
  • 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
  • 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
  • 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, DOI 10.17323/1609-4514-2004-4-1-217-244
  • 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
  • Graeme Segal, Categories and cohomology theories, Topology 13 (1974), 293–312. MR 353298, DOI 10.1016/0040-9383(74)90022-6
  • 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
  • S. Takagi: Compactifying $spec {\mathbb Z}$, preprint, arXiv: mathAG/1203.4914.
  • B. Toën, M. Vaquié: Au-dessous de $spec {\mathbb Z}$, preprint, arXiv: math/0509684.
  • A. Weil: Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques [$1939_a$], Oevres Scient. I, 236-240, (1980), Springer-Verlag; Revue Scientifique 77 (1939) 104-106.