Morphisms, products and fibre products of algebraic spaces

Morphisms of algebraic spaces over $S$ are the natural transformations of their underlying presheaves (Algebraic spaces over a scheme, defined as fppf sheaves); the category of algebraic spaces over $S$ is a full subcategory of the presheaves on $(Sch\mathrm{/}S{)}_{fppf}$, so a morphism is a morphism of sheaves and composition is composition of natural transformations.

For algebraic spaces $F,G,H$ and morphisms $F\to H$, $G\to H$, the fibre product of presheaves $F{\times }_{H}G$ is computed objectwise by $(F{\times }_{H}G)(T)=F(T){\times }_{H(T)}G(T)$ (Fibre product of schemes, Presheaves, covariantly and contravariantly representable functors, and representations); it is again an algebraic space over $S$ and represents the fibre product in the category of algebraic spaces. Indeed $F{\times }_{H}G$ is an fppf sheaf, since limits of sheaves are computed objectwise. For a scheme $T$ and two sections of $F{\times }_{H}G$ over $T$, their equality locus is the fibre product over $T$ of the scheme-valued equality loci of their $F$- and $G$-components. These loci are schemes by representability of ${\mathrm{\Delta }}_F$ and ${\mathrm{\Delta }}_G$, so the diagonal of $F{\times }_{H}G$ is representable. Choose etale scheme covers $U_F\to F$ and $U_G\to G$. The sheaf $W=U_F{\times }_H{U}_G$ is a scheme: it is the pullback of the representable diagonal ${\mathrm{\Delta }}_H$ along $U_F{\times }_S{U}_G\to H\times H$. The map $W\to F{\times }_{H}G$ is representable, etale and surjective. To check this on a scheme $T\to F{\times }_{H}G$, its base change is the product over $T$ of the etale surjective schemes $T{\times }_F{U}_F$ and $T{\times }_G{U}_G$; their product is etale and surjective over $T$. Thus $W$ is the required cover. These constructions use only scheme fibre products and the three given diagonals; no lifts to a chosen cover of $H$ are required. In particular products $F{\times }_{S}G$ over the terminal algebraic space $S$ are algebraic spaces over $S$. The diagonal ${\mathrm{\Delta }}_F\text{\hspace{0.17em}⁣}:F\to F{\times }_{S}F$ is a morphism of algebraic spaces representable by schemes, by condition 2 of Algebraic spaces over a scheme, defined as fppf sheaves.

Let $\mathcal{P}$ be a property of scheme morphisms stable under base change. For a morphism $f\text{\hspace{0.17em}⁣}:F\to G$ representable by schemes, the representable property $\mathcal{P}$ means that every base change $F{\times }_{G}T\to T$ to a scheme has $\mathcal{P}$ (Representable morphisms of presheaves and fibrewise properties). In particular, an open immersion is a morphism representable by open immersions (Open immersions of schemes). A morphism $f$ is separated when its relative diagonal ${\mathrm{\Delta }}_f\text{\hspace{0.17em}⁣}:F\to F{\times }_{G}F$ is a closed immersion (Closed immersions of schemes); this diagonal is representable by schemes, being a base change of ${\mathrm{\Delta }}_F$.

If $\mathcal{P}$ is local in the etale topology on both source and target, it extends to arbitrary morphisms of algebraic spaces by scheme charts: $f\text{\hspace{0.17em}⁣}:F\to G$ has property $\mathcal{P}$ when for every commutative square with top arrow $h\text{\hspace{0.17em}⁣}:U\to V$, bottom arrow $f$, and representable etale vertical arrows $U\to F$ and $V\to G$ from schemes, $h$ has $\mathcal{P}$. Thus $f$ is etale when these scheme morphisms $h$ are etale (Étale morphism of schemes). For properties additionally stable under base change and fppf-local on the target, this chart definition agrees with the preceding fibrewise definition whenever $f$ is representable by schemes. In particular representable etale requires both scheme representability and etaleness; a general etale morphism of algebraic spaces need not be representable by schemes.