To begin, we must first specify what a “local” ring is. Let $\StrTop{\ETop}$ be a topos ringed in $k$-algebras. It [trans.: $\Str{\ETop}$] can be identified with a left exact functor $M\to \ETop$ that sends an affine variety $V$ to the object $V\prn{\Str{\ETop}}$ of $\Str{\ETop}$-points of $V$. $\StrTop{\ETop}$ will be “local” if the functor is continuous for the topology on $M$, i.e. if whenever $\prn{U_i\to V}\Sub{i\in I}$ is a covering, the family

is surjective. In the case of the Zariski topology, this gives the local rings (without quotation marks) and in the case of the étale topology it gives the strict local rings (or Henselian local rings with a separably closed residue field).

$\Sh{M}$ is the classifying topos of “local” $k$-algebras, which is to say that the data of a topos $\StrTop{\ETop}$ ringed in “local” $k$-algebras amounts to that of a geometric morphism $\ETop\to\Sh{M}$ whose inverse image functor associates to an object $X$ of $\Sh{M}$ the object $X\prn{\Str{\ETop}}$ of $\Str{\ETop}$-points of $X$. Note that to give an $\Str{\ETop}$-point $p:1\to X\prn{\Str{\ETop}}$ of $X$ amounts to giving a factorization

Let $\StrTop{\ETop}$ and $\StrTop{\FTop}$ be two topoi ringed in local $k$-algebras. A morphism from $\StrTop{\ETop}$ to $\StrTop{\FTop}$ is a geometric morphism $\phi : \ETop\to \FTop$ together with a morphism of $k$-algebras $f : \InvImg{\phi}\Str{\FTop}\to \Str{\ETop}$. This morphism will be “local” if for each elementary étalé $U\to V$ the square

is cartesian. We thus recover the local morphisms (without quotation marks) for the two localization data already considered.

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 $\prn{\Sh{\ET/Y},\Str{Y}}$.