Petits et gros topos en géométrie algébrique

We will fix in $M$ a set $\ET$ of morphisms that we want to think of like local isomorphisms. In the examples that we shall consider, these will be either open immersions or étale morphisms. The morphisms of $\ET$ shall be called elementary étalés. We assume that they satisfy the following properties, which are natural given how we think of them, and satisfied by both open immersions and étale morphisms.

1. Every isomorphism is an elementary étalé.
2. The composition of two elementary étalés is an elementary étalé.
3. The elementary étalés are stable under change of base.
4. If both $u\circ v$ and $v$ are elementary étalés, then so is $u$.

In the terminology of Bénabou, $\ET$ is a calibration in $M$.

The elementary étalés are used to make coverings: we will associate to each affine variety $V$ a set of families of elementary étalés over $V$ that are the covers of $V$. We conventionally ask that isomorphisms are covering, and that coverings are stable under change of base and that they have local character.

We are then in possession of a localization datum on $M$. A localization datum is therefore nothing more than a Grothendieck topology on $M$ in which we have specified the morphisms that are used to make the covers. In algebraic geometry there are well-known localization datums: the Zariski topology with open immersions, and the étale topology with étale morphisms.

Let $\Sh{M}$ be the topos of sheaves on $M$ with the topology of the localization datum. We denote by $\Yo : M \to \Sh{M}$ the canonical functor. The objects of $\Sh{M}$ are obtained by gluing of affine varieties; the method of gluing is specified by the topology on $M$.

Let us now choose an affine variety $V$. Let $\ET/V$ be the full subcategory of $M/V$ spanned elementary étalés over $V$. $\ET/V$ inherits a Grothendieck topology from the localization datum. Let $\Sh{\ET/V}$ be the topos of sheaves for this topology. The objects of $\Sh{\ET/V}$ are obtained by gluing elementary étalés over $V$; it is thus natural to think of them as the étalés over $V$.

The immersion $i:\ET/V\hookrightarrow M/V$ is left exact, continuous and cocontinuous (Artin et al., 1972). It extends into an immersion

which will allow us to identify the étalés over $V$ with objects of $\Sh{M}$ over $V$. $\Sum{i}$ is left exact, and has a right adjoint $i^\sharp$ (the associated “étalé” functor) which itself has a right adjoint $\Prod{i}$. We therefore have two geometric morphisms

and the composite $i^\sharp\circ\Sum{i}$ is isomorphic to the identity.

The relation between $\Sh{\ET/V}$ and $\Sh{M}/\Yo{V}$ is the same as the one between the little topos and the big topos of a topological space (Artin et al., 1972, sec. IV.4.10). There is also in SGA4 an exercise whose content is more or less what we have just said. (Puzzle: find a point in common between La Samaritaine and SGA 4.)

We know what the étalés on an affine variety are. The étalés on any object $X$ of $\Sh{M}$ are defined to be the morphisms $Y\to X$ whose inverse image by each morphism $\Yo{V}\to X$ is étalé over $V$. The étalés have properties 1–4 of étalés elementaires. They also obtain a local character: if $\prn{X_i\to X}\Sub{j\in J}$ is a covering family, $Y\to X$ is étalé when each $Y\times_X X_j\to X_j$ is.

We now come to the notion of a variety with respect to a localization datum. A variety is an object $X$ of $\Sh{M}$ that is covered by affine varieties that are étalé over $X$. The advantage of varieties in comparison to other objects of $\Sh{M}$ is: if $X$ is a variety, let $\ET/X$ be the full subcategory of $M/X$ spanned by affine étalés over $X$. $\Sh{\ET/X}$ is the topos of étale’s over $X$ (we may remark that if $Y$ is étalé over a variety $X$, then $Y$ is also a variety). We have the same functors $\Sum{i}$, $i^\sharp$, and $\Prod{i}$ between the little topos $\Sh{\ET/X}$ and the big topos $\Sh{M}/X$.

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