Wikipedia's definition of an Algebraic Space

Question

Wikipedia defines an algebraic space $\mathfrak{X}$ to be a sheaf on the big étale site $(\text{Sch}/S)_{et}$, such that:

1. There is a surjective étale morphism $h_X\to \mathfrak{X}$
2. The diagonal morphism $\Delta_{\mathfrak{X}/S}:\mathfrak{X}\to\mathfrak{X}\times\mathfrak{X}$ is representable.

What does the first condition actually mean? Possibly it makes sense by means of the second condition, but I wasn't sure. What does it mean for a morphism onto an étale sheaf to be étale?

Answer 1

$\require{AMScd}$tl;dr: The first condition only makes sense in light of the latter. By definition, if $P$ is a stable property, a representable morphism $f : F\to G$ of étale sheaves has property $P$ if the base change of $f$ along any morphism $T\to G$ has property $P,$ where $T$ is a scheme. Surjectivity and étaleness are stable properties, so we may talk about a representable morphism of sheaves being étale surjective.

---

Here are the details. Everything I'm saying below can be found in the book "Algebraic Spaces and Stacks" by Martin Olsson. In particular, this is the discussion at the start of chapter 5.

Essentially, the idea is that certain properties of morphisms of schemes have nice local and base-change compatibility properties that allow us to talk about that property for general sheaves.

Let's fix a scheme $S,$ and suppose that we have a morphism of sheaves $f : F\to G$ on $(\mathsf{Sch}_S)_{et}$ which is representable by schemes. If $P$ is a property of morphisms of schemes, we want to define $f$ to have property $P$ if the base change of $f$ along any morphism $T\to G$ has property $P,$ where $T$ is an $S$-scheme (viewed as a representable sheaf).

That is, we say that $f$ has property $P$ if for every $S$-scheme $T$ and morphism $T\to G$, the morphism (of schemes!) $\operatorname{pr}_2 : F\times_G T\to T$ has property $P.$

Of course, there's a potential issue here. We want to be sure that if we start with a morphism of schemes $f : X\to Y,$ that the induced morphism $h_f : h_X\to h_Y$ has property $P$ in the above sense if and only if $f$ has $P$ in the classical sense.

Let's see what we might need for this to be true. Let $T$ be an $S$-scheme and $g : T\to Y$ be a morphism of schemes. (Or equivalently, let $g$ be a morphism of étale sheaves. Indeed, if $X$ and $Y$ are schemes, any morphism $h_X\to h_Y$ is representable by schemes.) Then we have $h_T\times_{h_Y} h_X\simeq h_{X\times_Y T},$ and this isomorphism is compatible with the projection morphisms. So, $h_f$ has property $P$ in the above sense if and only if for every morphism of schemes $g : T\to Y$ the projection $\operatorname{pr}_2 : X\times_Y T\to T$ has property $P.$

It turns out that the right condition to ask for is that $P$ is stable. Essentially, this amounts to $P$ containing isomorphisms, being a property which is closed under composition and under base change, and which plays nicely with the étale topology. I'll define stability below for any site.

Definition 5.1.3: Let $C$ be a site.

(i) A closed subcategory of $C$ is a subcategory $D\subseteq C$ such that $D$ contains all isomorphisms, and for all Cartesian diagrams in $C$ \begin{CD} X' @>>> X \\ @VVV \lrcorner @VVV\\ Y' @>>> Y \end{CD} for which the morphism $X\to Y$ is in $D,$ we have $X'\to Y'$ in $D.$

(ii) A closed subcategory $D\subseteq C$ is stable if for all $f : X\to Y$ in $C$ and every covering $\{Y_i\to Y\},$ the morphism $f$ is in $D$ if and only if all the maps $f_i : X\times_Y Y_i\to Y_i$ are in $D.$

(iv) If $P$ is a property of morphisms in $C$ satisfied by isomorphisms and closed under composition, let $D_P$ be the subcategory of $C$ with the same objects as $C,$ but whose morphisms are morphisms satisfying $P.$ We say that $P$ is stable if the subcategory $D_P\subseteq C$ is stable.

And now, one can verify that if $P$ is stable, a morphism of schemes $f : X\to Y$ has property $P$ if and only if $h_f : h_X\to h_Y$ has property $P.$ Additionally, we have the following crucial proposition.

Proposition 5.1.4: Let $S$ be a scheme, and let $C$ be the category of $S$-schemes with the étale topology.

(i) The following properties of morphisms in $C$ are stable: proper, separated, surjective, quasi-compact.

(ii) The following properties of morphisms in $C$ are stable (and local on domain): locally of finite type, locally of finite presentation, flat, étale, universally open, locally quasi-finite, smooth.

So, surjectivity and étaleness are both properties one can talk about in this sense. Thus, it makes sense to talk about a morphism $h_X\to\mathfrak{X}$ being surjective and étale. Or at least, it almost does. First, we need a lemma.

Lemma 5.1.9 Let $S$ be a scheme and let $F$ be a sheaf on $(\mathsf{Sch}_S)_{et}.$ Suppose that the diagonal morphism $\Delta : F\to F\times F$ is representable by schemes. Then if $T$ is a scheme, any morphism $f : T\to F$ is representable by schemes.

Proof: If $T, T'$ are schemes, the fiber product of \begin{CD} &&T \\ && @VVV\\ T' @>>> F \end{CD} is isomorphic to the fiber product of \begin{CD} &&T\times_S T' \\ && @VVV\\ F @>>> F\times F. \end{CD} Since $\Delta$ is representable by schemes, so is this fiber product.