nLab Spec(Z)

Idea

$Spec(\mathbb{Z})$ denotes the spectrum of the commutative ring $\mathbb{Z}$ of integers. Its underlying topological space (also known as the “prime spectrum” or “Zariski spectrum” of $\mathbb{Z}$) has the prime ideals of $\mathbb{Z}$ as points and carries the Zariski topology. The closed points are the maximal ideals $(p)$, for each prime number $p$ in $\mathbb{Z}$; the non-maximal prime ideal $(0)$ is a generic point as it has as closure the whole of $Spec(\mathbb{Z})$. A subset of $Spec(\mathbb{Z})$ is closed if and only if it is all of $Spec(\mathbb{Z})$ or consists of finitely many points of the form $(p)$ with $p$ prime.

As a ringed space, $Spec(\mathbb{Z})$ carries a structure sheaf $\mathcal{O}$ which is a sheaf of commutative rings on the Zariski spectrum of $\mathbb{Z}$. Given a typical open set

$$U = Spec(\mathbb{Z})\setminus \{(p_1),\ldots,(p_n)\}$$

we have

$$\mathcal{O}(U) = \left\{ \frac{a}{b}\in\mathbb{Q}\mid a,b\in\mathbb{Z}\ \text{and all primes dividing}\ b\ \text{are among}\ p_1,\ldots,p_n \right\}.$$

The restriction morphisms are the natural inclusions.

The stalk at the point $(p)$ is the local ring $\mathbb{Z}_{(p)}=\left\{\frac{a}{b}\in\mathbb{Q}\mid a\in\mathbb{Z},\ b\in\mathbb{Z}\setminus (p)\right\}$ with residue field $\mathbb{F}_p$. The stalk at $(0)$ is $\mathbb{Q}$.

The idea here is that a rational number $a/b$ with coprime integers $a,b$ acts like a rational function on the prime spectrum $Spec(\mathbb{Z})$ (which we think of as analogous to a complex variety, say), having a pole at $(p)$ whenever the prime $p$ divides $b$, and assigning the value $(a+(p))/(b+(p))\in\mathbb{F}_p$ to $(p)$ when $p$ does not divide $b$. In this view, the structure sheaf $\mathcal{O}$ is the sheaf of regular functions, i.e. $\mathcal{O}(U)$ consists of those “functions” that have no poles in $U$.

With these definitions, $Spec(\mathbb{Z})$ becomes a locally ringed space and an affine scheme.

Since $\mathbb{Z}$ is the initial object in the category CRing of commutative rings, $Spec(\mathbb{Z})$ is the terminal object in the category of schemes.

The functor of points corresponding to $Spec(\mathbb{Z})$ is the constant functor that assigns a singleton to every commutative ring. (see functorial geometry).

The gros etale topos over $Spec(\mathbb{Z})$ is the context for arithmetic geometry. By the discussion at Borger's absolute geometry it sits via an essential geometric morphism over the $\mathbb{F}_1$-topos:

$$Et\big(Spec(\mathbb{Z})\big)\longrightarrow Et\big(Spec(\mathbb{F}_1)\big) \mathrlap{\,.}$$

Properties

There are some phenomena that may be interpreted as $Spec(\mathbb{Z})$ behaving like a 3-manifold in some ways.

As a 3-sphere containing knots

Several properties of $Spec(\mathbb{Z})$ make it behave as if of dimension 3. For instance $Spec(\mathbb{Z}) \cup \{\infty\}$ has étale cohomological dimension equal to 3, up to 2-torsion (Mazur 73). Moreover the étale fundamental group $\hat \pi_1(Spec(\mathbb{Z}) \cup \{\infty\})$ is trivial, and hence Mazur suggested that $Spec(\mathbb{Z}) \cup \{\infty\}$ is in fact analogous to the 3-sphere.

Similarly, the spectra $Spec(\mathbb{F}_q)$ of finite fields look like compact 1-dimensional spaces – circles – in that their étale cohomology with $\mathbb{Z}_l$-coefficients for $l$ coprime to $q$ is $\mathbb{Z}_l$ in degrees 0 and 1 and vanishes in all higher degrees.

From this it is folklore (going back to Mazur and Manin, review includes Deninger 05, section 8, Kohno-Morishita 06) that the spectra of prime fields with their canonical embedding into $Spec(\mathbb{Z})$

$$Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$$

(formally dual to the canonical mod-$p$ projection $\mathbb{Z}\to \mathbb{F}_p$) are analogous to knots inside this 3-dimensional space (a good exposition is in LeBruyn).

Observations like this give rise to the field of arithmetic topology.

As a hyperbolic 3-manifold containing prime geodesics

However, in view of the analogy between the Selberg zeta function and the Artin L-function it might be more appropriate to think of $Spec(\mathbb{Z})$ as analogous to a hyperbolic manifold of dimension 3 (see also Fujiwara 07, slide 7) and then to think of finite field spectra as analogous to the prime geodesics in the manifold. This does not change the fact that every single $Spec(\mathbb{F}_p) \hookrightarrow Spec(\mathbb{Z})$ is like an embedded circle, hence like a knot, but it affects the perspective on which role these play. For instance there does not seem to be a differential geometric analog situation where one considers infinite products over all knots in a 3-space, but there are such situations where one considers infinite products over all prime geodesics in a space, namely the Selberg zeta function analogous to the Artin L-function with its product over prime ideals.

Function field analogy

function field analogy

	number fields (“function fields of curves over F1”)	function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves)	Riemann surfaces/complex curves
affine and projective line
	$\mathbb{Z}$ (integers)	$\mathbb{F}_q[z]$ (polynomials, polynomial algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	$\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane)
	$\mathbb{Q}$ (rational numbers)	$\mathbb{F}_q(z)$ (rational fractions/rational function on affine line $\mathbb{A}^1_{\mathbb{F}_q}$)	meromorphic functions on complex plane
	$p$ (prime number/non-archimedean place)	$x \in \mathbb{F}_q$, where $z - x \in \mathbb{F}_q[z]$ is the irreducible monic polynomial of degree one	$x \in \mathbb{C}$, where $z - x \in \mathcal{O}_{\mathbb{C}}$ is the function which subtracts the complex number $x$ from the variable $z$
	$\infty$ (place at infinity)		$\infty$
	$Spec(\mathbb{Z})$	$\mathbb{A}^1_{\mathbb{F}_q}$ (affine line)	complex plane
	$Spec(\mathbb{Z}) \cup place_{\infty}$	$\mathbb{P}_{\mathbb{F}_q}$ (projective line)	Riemann sphere
	$\partial_p \coloneqq \frac{(-)^p - (-)}{p}$ (Fermat quotient)	$\frac{\partial}{\partial z}$ (coordinate derivation)	“
	genus of the rational numbers = 0		genus of the Riemann sphere = 0
formal neighbourhoods
	$\mathbb{Z}/(p^n \mathbb{Z})$ (prime power local ring)	$\mathbb{F}_q [z]/\big((z-x)^n \mathbb{F}_q [z]\big)$ ($n$-th order univariate local Artinian $\mathbb{F}_q$-algebra)	$\mathbb{C}[z]/\big((z-x)^n \mathbb{C}[z]\big)$ ($n$-th order univariate Weil $\mathbb{C}$-algebra)
	$\mathbb{Z}_p$ (p-adic integers)	$\mathbb{F}_q[ [ z -x ] ]$ (power series around $x$)	$\mathbb{C}[ [z-x] ]$ (holomorphic functions on formal disk around $x$)
	$Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X$ (“$p$-arithmetic jet space” of $X$ at $p$)		formal disks in $X$
	$\mathbb{Q}_p$ (p-adic numbers)	$\mathbb{F}_q((z-x))$ (Laurent series around $x$)	$\mathbb{C}((z-x))$ (holomorphic functions on punctured formal disk around $x$)
	$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles)	$\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
	$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles)	$\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field )	$\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))$
theta functions
	Jacobi theta function
zeta functions
	Riemann zeta function	Goss zeta function
branched covering curves
	$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension)	$K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_q$	$K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$)
	$\mathcal{O}_K$ (ring of integers)		$\mathcal{O}_{\Sigma}$ (structure sheaf)
	$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places)	$\Sigma$ (arithmetic curve)	$\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere)
	$\frac{(-)^p - \Phi(-)}{p}$ (lift of Frobenius morphism/Lambda-ring structure)	$\frac{\partial}{\partial z}$	“
	genus of a number field	genus of an algebraic curve	genus of a surface
formal neighbourhoods
	$v$ prime ideal in ring of integers $\mathcal{O}_K$	$x \in \Sigma$	$x \in \Sigma$
	$K_v$ (formal completion at $v$)		$\mathbb{C}((z_x))$ (function algebra on punctured formal disk around $x$)
	$\mathcal{O}_{K_v}$ (ring of integers of formal completion)		$\mathbb{C}[ [ z_x ] ]$ (function algebra on formal disk around $x$)
	$\mathbb{A}_K$ (ring of adeles)		$\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$)
	$\mathcal{O}$		$\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ]$ (function ring on all formal disks around all points in $\Sigma$)
	$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles)		$\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))$
Galois theory
	Galois group	“	$\pi_1(\Sigma)$ fundamental group
	Galois representation	“	flat connection (“local system”) on $\Sigma$
class field theory
	class field theory	“	geometric class field theory
	Hilbert reciprocity law	Artin reciprocity law	Weil reciprocity law
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)$ (idele class group)	“
	$GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})$	“	$Bun_{GL_1}(\Sigma)$ (moduli stack of line bundles, by Weil uniformization theorem)
non-abelian class field theory and automorphy
	number field Langlands correspondence	function field Langlands correspondence	geometric Langlands correspondence
	$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations)	“	$Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$, by Weil uniformization theorem)
	Tamagawa-Weil for number fields	Tamagawa-Weil for function fields
theta functions
	Hecke theta function		functional determinant line bundle of Dirac operator/chiral Laplace operator on $\Sigma$
zeta functions
	Dedekind zeta function	Weil zeta function	zeta function of a Riemann surface/of the Laplace operator on $\Sigma$
higher dimensional spaces
zeta functions	Hasse-Weil zeta function

Related concepts

• Spec(S)

• field with one element

References

• Barry Mazur, Notes on étale cohomology of number fields, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 6 no. 4 (1973), p. 521-552 (NUMDAM)

• Barry Mazur, Remarks on the Alexander polynomial, unpublished note

• Christopher Deninger, Arithmetic Geometry and Analysis on Foliated Spaces (arXiv:math/0505354)

• Toshitake Kohno, Masanori Morishita (eds.), Primes and Knots, Contemporary Mathematics, AMS 2006 (web)

• Lieven LeBruyn, talk 2010 (pdf slides (35 mb), MO comment (with more details))

• K. Fujiwara, $p$-adic gauge theory in number theory, 2007 (pdf slides)

• Bertrand Toën, Michel Vaquié, Au-dessous de $Spec \mathbb{Z}$, Journal of K-Theory 3 3 (2009) 437-500 [doi:10.1017/is008004027jkt048]