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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)
Published electronically: December 6, 2016
MSC: Primary 11G99, 11R99, 14A20, 14G40

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Table of Contents


  • 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


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 $\epsilon : \partial F\rightarrow \{\pm 1\}$.

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