Connes-Consani for undergraduates (3)

A quick recap of last time. We are trying to make sense of affine varieties over the elusive field with one element ${\mathbb{F}}_1$, which by Grothendieck’s scheme-philosophy should determine a functor

$\mathbf{nano}(N):\mathbf{abelian}\to \mathbf{sets}A\mapsto N(A)$

from finite Abelian groups to sets, typically giving pretty small sets $N(A)$. Using the F_un mantra that $\mathbb{Z}$ should be an algebra over ${\mathbb{F}}_1$ any ${\mathbb{F}}_1$-variety determines an integral scheme by extension of scalars, as well as a complex variety (by extending further to $\mathbb{C}$). We have already connected the complex variety with the original functor into a gadget that is a couple $(\mathbf{nano}(N),\mathbf{maxi}(R))$ where $R$ is the coordinate ring of a complex affine variety $X_R$ having the property that every element of $N(A)$ can be realized as a $\mathbb{C}A$-point of $X_R$. Ringtheoretically this simply means that to every element $x\in N(A)$ there is an algebra map $N_x:R\to \mathbb{C}A$.

Today we will determine which gadgets determine an integral scheme, and do so uniquely, and call them the sought for affine schemes over ${\mathbb{F}}_1$.

Let’s begin with our example : $\mathbf{nano}(N)={\underset{―}{\mathbb{G}}}_m$ being the forgetful functor, that is $N(A)=A$ for every finite Abelian group, then the complex algebra $R=\mathbb{C}[x,x^{-1}]$ partners up to form a gadget because to every element $a\in N(A)=A$ there is a natural algebra map $N_a:\mathbb{C}[x,x^{-1}]\to \mathbb{C}A$ defined by sending $x\mapsto e_a$. Clearly, there is an obvious integral form of this complex algebra, namely $\mathbb{Z}[x,x^{-1}]$ but we have already seen that this algebra represents the mini-functor

$\mathbf{min}(\mathbb{Z}[x,x^{-1}]):\mathbf{abelian}\to \mathbf{sets}A\mapsto (\mathbb{Z}A{)}^{\ast }$

and that the group of units $(\mathbb{Z}A{)}^{\ast }$ of the integral group ring $\mathbb{Z}A$ usually is a lot bigger than $N(A)=A$. So, perhaps there is another less obvious $\mathbb{Z}$-algebra $S$ doing a much better job at approximating $N$? That is, if we can formulate this more precisely…

In general, every $\mathbb{Z}$-algebra $S$ defines a gadget $\mathbf{gadget}(S)=(\mathbf{mini}(S),\mathbf{maxi}(S{\otimes }_{\mathbb{Z}}\mathbb{C}))$ with the obvious (that is, extension of scalars) evaluation map

$\mathbf{mini}(S)(A)=Ho{m}_{\mathbb{Z}-alg}(S,\mathbb{Z}A)\to Ho{m}_{\mathbb{C}-alg}(S{\otimes }_{\mathbb{Z}}\mathbb{C},\mathbb{C}A)=\mathbf{maxi}(S{\otimes }_{\mathbb{Z}}\mathbb{C})(A)$

Right, so how might one express the fact that the integral affine scheme $X_T$ with integral algebra $T$ is the ‘best’ integral approximation of a gadget $(\mathbf{nano}(N),\mathbf{maxi}(R))$. Well, to begin its representing functor should at least contain the information given by $N$, that is, $\mathbf{nano}(N)$ is a sub-functor of $\mathbf{mini}(T)$ (meaning that for every finite Abelian group $A$ we have a natural inclusion $N(A)\subset Ho{m}_{\mathbb{Z}-alg}(T,\mathbb{Z}A)$). As to the “best”-part, we must express that all other candidates factor through $T$. That is, suppose we have an integral algebra $S$ and a morphism of gadgets (as defined last time)

$f:(\mathbf{nano}(N),\mathbf{maxi}(R))\to \mathbf{gadget}(S)=(\mathbf{mini}(S),\mathbf{maxi}(S{\otimes }_{\mathbb{Z}}\mathbb{C}))$

then there ought to be $\mathbb{Z}$-algebra morphism $T\to S$ such that the above map $f$ factors through an induced gadget-map $\mathbf{gadget}(T)\to \mathbf{gadget}(S)$.

Fine, but is this definition good enough in our trivial example? In other words, is the “obvious” integral ring $\mathbb{Z}[x,x^{-1}]$ the best integral choice for approximating the forgetful functor $N={\underset{―}{\mathbb{G}}}_m$? Well, take any finitely generated integral algebra $S$, then saying that there is a morphism of gadgets from $({\underset{―}{\mathbb{G}}}_m,\mathbf{maxi}(\mathbb{C}[x,x^{-1}])$ to $\mathbf{gadget}(S)$ means that there is a $\mathbb{C}$-algebra map $\psi :S{\otimes }_{\mathbb{Z}}\mathbb{C}\to \mathbb{C}[x,x^{-1}]$ such that for every finite Abelian group $A$ we have a commuting diagram

Here, $e$ is the natural evaluation map defined before sending a group-element $a\in A$ to the algebra map defined by $x\mapsto e_a$ and the vertical map on the right-hand side is extensions by scalars. From this data we must be able to show that the image of the algebra map

is contained in the integral subalgebra $\mathbb{Z}[x,x^{-1}]$. So, take any generator $z$ of $S$ then its image $\psi (z)\in \mathbb{C}[x,x^{-1}]$ is a Laurent polynomial of degree say $d$ (that is, $\psi (z)=c_{-d}{x}^{-d}+\dots c_{-1}{x}^{-1}+c_0+c_{1}x+\dots +c_d{x}^d$ with all coefficients a priori in $\mathbb{C}$ and we need to talk them into $\mathbb{Z}$).

Now comes the basic trick : take a cyclic group $A=C_N$ of order $N>d$, then the above commuting diagram applied to the generator of $C_N$ (the evaluation of which is the natural projection map $\pi :\mathbb{C}[x.x^{-1}]\to \mathbb{C}[x,x^{-1}]/(x^N-1)=\mathbb{C}{C}_N$) gives us the commuting diagram

where the horizontal map $j$ is the natural inclusion map. Tracing $z\in S$ along the diagram we see that indeed all coefficients of $\psi (z)$ have to be integers! Applying the same argument to the other generators of $S$ (possibly for varying values of N) we see that , indeed, $\psi (S)\subset \mathbb{Z}[x,x^{-1}]$ and hence that $\mathbb{Z}[x,x^{-1}]$ is the best integral approximation for ${\underset{―}{\mathbb{G}}}_m$.

That is, we have our first example of an affine variety over the field with one element ${\mathbb{F}}_1$ : $({\underset{―}{\mathbb{G}}}_m,\mathbf{maxi}(\mathbb{C}[x,x^{-1}])\to \mathbf{gadget}(\mathbb{Z}[x,x^{-1}])$.

What makes this example work is that the infinite group $\mathbb{Z}$ (of which the complex group-algebra is the algebra $\mathbb{C}[x,x^{-1}]$) has enough finite Abelian group-quotients. In other words, ${\mathbb{F}}_1$ doesn’t see $\mathbb{Z}$ but rather its profinite completion $\text{Missing argument for mathbb}$… (to be continued when we’ll consider noncommutative ${\mathbb{F}}_1$-schemes)

In general, an affine ${\mathbb{F}}_1$-scheme is a gadget with morphism of gadgets
$(\mathbf{nano}(N),\mathbf{maxi}(R))\to \mathbf{gadget}(S)$ provided that the integral algebra $S$ is the best integral approximation in the sense made explicit before. This rounds up our first attempt to understand the Connes-Consani approach to define geometry over ${\mathbb{F}}_1$ apart from one important omission : we have only considered functors to $\mathbf{sets}$, whereas it is crucial in the Connes-Consani paper to consider more generally functors to graded sets. In the final part of this series we’ll explain what that’s all about.