[Paper Review] Compactifying Spec Z

Satoshi Takagi|arXiv (Cornell University)|Mar 22, 2012

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper introduces convexoid rings—algebraic structures generalizing rings by relaxing associativity of addition—to provide a purely algebraic construction of the compactified spectrum $ύbar{\operatorname{Spec}\mathbb{Z}} = \operatorname{Spec}\mathbb{Z} \cup \{\infty\}$, realized as the Zariski-Riemann space of $\operatorname{Spec}\mathbb{Z}$ in the category of weak convexoid schemes. The key result is a canonical, universal definition of the infinite place via the valuation convexoid ring $\mathbf{D}\mathbb{Q}$, embedding the archimedean norm algebraically without ad hoc assumptions.

ABSTRACT

In this paper, we introduce a new algebraic type of `convexoid rings', and we give the definition of (weak) convexoid schemes, which share similar properties with ordinary schemes. As a result, we give a purely-algebraic construction of the compactification of Spec Z (in Arakelov's sense) which is realized as the Zariski-Riemann space in the category of weak convexoid schemes.

Motivation & Objective

• To resolve the longstanding problem of constructing the compactification $\overline{\operatorname{Spec}\mathbb{Z}} = \operatorname{Spec}\mathbb{Z} \cup \{\infty\}$ in a canonical, algebraic way, avoiding ad hoc extensions.
• To address the limitation of ordinary schemes, where the infinite place $\infty$ cannot be realized due to $\operatorname{Spec}\mathbb{Z}$ being a final object.
• To provide a universal, intrinsic definition of the archimedean place by deriving the norm structure from algebraic axioms rather than external data.
• To generalize Arakelov geometry and $\mathbb{F}_1$-geometry by embedding the infinite place within a broader algebraic framework.
• To establish a correspondence between the global sections of $\overline{\operatorname{Spec}\mathbb{Z}}$ and $\mathbb{F}_{1^2} = \{0, \pm 1\}$, aligning with $\mathbb{F}_1$-geometric expectations.

Proposed method

• Introduce convexoid rings as commutative monoids under multiplication with a non-associative addition $\boxplus$, generalizing rings and multiplicative monoids with absorbing elements.
• Define weak convexoid schemes to allow for 'twisted' patching, enabling the treatment of Zariski-Riemann spaces as legitimate geometric objects.
• Construct $\operatorname{Proj}R_0$ for the initial convexoid ring $R_0$, which serves as the 'fake closure' of $\operatorname{Spec}\mathbb{Z}$ with underlying space homeomorphic to $\overline{\operatorname{Spec}\mathbb{Z}}$.
• Use graded convexoid rings to define projective convexoid schemes analogously to classical algebraic geometry, with $\mathbb{F}_{1^2}$-modules replacing vector spaces.
• Define a morphism $f: \operatorname{Proj}R_0 \to \mathbb{P}$ to a proprojective space over $\mathbb{F}_{1^2}$, which becomes an immersion after taking the infinite product of projective spaces.
• Leverage a variant of Ostrowski’s theorem formulated entirely in algebraic terms to justify the universal property of the infinite place.

Experimental results

Research questions

• RQ1 Can the infinite place $\infty$ be constructed within the category of schemes using only algebraic structures?
• RQ2 Is there a universal, intrinsic algebraic definition of the archimedean norm on $\mathbb{Q}$ that does not rely on analytic data?
• RQ3 How can the Zariski-Riemann space of $\operatorname{Spec}\mathbb{Z}$ be realized as a geometric object in a generalized scheme theory?
• RQ4 What is the role of $\mathbb{F}_{1^2}$ in the global sections of $\overline{\operatorname{Spec}\mathbb{Z}}$, and how does it relate to $\mathbb{F}_1$-geometry?
• RQ5 Can the compactification $\overline{\operatorname{Spec}\mathbb{Z}}$ be defined via a universal property in a category of generalized schemes?

Key findings

• The compactification $\overline{\operatorname{Spec}\mathbb{Z}} = \operatorname{Spec}\mathbb{Z} \cup \{\infty\}$ is canonically realized as the Zariski-Riemann space of $\operatorname{Spec}\mathbb{Z}$ in the category of weak convexoid schemes.
• The stalk at the infinite place $\infty$ is the valuation convexoid ring $\mathbf{D}\mathbb{Q}$, consisting of rational numbers with absolute value $\leq 1$, equipped with a non-associative addition $\boxplus$.
• The global sections $\Gamma(\overline{\operatorname{Spec}\mathbb{Z}}, \mathscr{O})$ form the multiplicative monoid $\{0, \pm 1\}$, identified as $\mathbb{F}_{1^2}$, with no $\boxplus$-structure.
• The construction is universal: $\overline{\operatorname{Spec}\mathbb{Z}}$ is the universal weak convexoid scheme mapping to $\operatorname{Proj}R_0$, where $R_0$ is the initial convexoid ring.
• The morphism $f: \operatorname{Proj}R_0 \to \mathbb{P} = \prod_d \mathbb{P}^{2^d - 1}_{\mathbb{F}_{1^2}}$ becomes an immersion, showing that the infinite place is algebraically distinguishable.
• The archimedean norm on $\mathbb{Q}$ is derived from the algebraic structure of $\mathbf{D}\mathbb{Q}$, making the construction intrinsic and canonical, unlike in Arakelov or Durov geometry.