We now come to the representation of a variety $X$ as a “locally” ringed topos. The greater part of the work is already done. We have constructed the topos $\Sh{\ET/X}$ and a geometric morphism

which gives the desired “local” $k$-algebra. Explanation: the generic “local” $k$-algebra in $\Sh{M}$ is $\Yo{\AffLine}$, the affine line with its structure sheaf. Its inverse image in $\Sh{M}/X$ is $\Yo{\AffLine}\times X\to X$. We will write $\Str{X}$ for the associated étalé over $X$. Let $Y$ be an étalé variety on $X$. The sections of $\Str{X}$ over $Y$ are exactly the morphisms from $Y$ to $\Yo{\AffLine}$ in the topos $\Sh{M}$, i.e. the “functions” on $Y$. Thus with a localization datum comes a notion of function and $\Str{X}$ must be seen as the sheaf of functions on $X$. For the two known localization data, the functions are the regular functions.

We have thus constructed from the variety $X$ a topos $\prn{\Sh{\ET/X},\Str{X}}$ ringed in “local” $k$-algebras. The following properties show that this is a good representation:

1. Let $\StrTop{\ETop}$ be a topos ringed in “local” $k$-algebras. There is a bijection (if we drop “up to isomorphism”) between $\Str{\ETop}$-points of $X$ and local morphisms from $\StrTop{\ETop}$ to $\prn{\Sh{\ET/X},\Str{X}}$. If $X=\Yo{V}$, an $\Str{\ETop}$-point of $X$ is a morphism of $k$-algebras from $k[V]$ to $\GSec\StrTop{\ETop}$ and we recognize here the universal property of the spectrum.

   It is happening like this: fix $p:1\to X\prn{\Str{\ETop}}$. We obtain a geometric morphism

   $$ \ETop\to\Sh{M}/X\to \Sh{\ET/X}\text{.} $$

   Let $\Str{X,p}$ be the inverse image of $\Str{X}$ under this morphism; $\Str{X,p}$ is the “local” ring of $X$ at $p$, and we have a “local” morphism $\Str{X,p}\to\Str{\ETop}$. The latter comes from the natural transformation $\Sum{i}\circ i^\sharp \Rightarrow \Idn$:

   To show that we have a bijection, it suffices to observe that if we have a “local” morphism $f:\Str{\ETop}\to \Str{\ETop}\tick$, an $\Str{\ETop}$-point $p$ of $X$ and its image $q$ under $f$, the local rings $\Str{X,p}$ and $\Str{X,q}$ are isomorphic:

   $f$ local entails $f\cdot\Sum{i}$ iso.

2. Let $Y$ be another variety. There is a bijection between the morphisms from $X$ to $Y$ in $\Sh{M}$ and the local morphisms from $\prn{\Sh{\ET/X},\Str{X}}$ to $\prn{\Sh{\ET/Y},\Str{Y}}$.