nLab Spec(Z)

Idea

$Spec(\mathbb{Z})$ denotes 𝕊 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

[[!include function field analogy – table]]

Related concepts

• $Spec(\mathbb{S})$

• 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]