The top arrow of a “magic diagram” of algebraic spaces has nice immersion-like properties, and under separatedness hypotheses these get stronger.
Lemma 67.4.5. Let S be a scheme. Let f : X \to Z, g : Y \to Z and Z \to T be morphisms of algebraic spaces over S. Consider the induced morphism i : X \times _ Z Y \to X \times _ T Y. Then
i is representable, locally of finite type, locally quasi-finite, separated and a monomorphism,
if Z \to T is locally separated, then i is an immersion,
if Z \to T is separated, then i is a closed immersion, and
if Z \to T is quasi-separated, then i is quasi-compact.
Proof.
By general category theory the following diagram
\xymatrix{ X \times _ Z Y \ar[r]_ i \ar[d] & X \times _ T Y \ar[d] \\ Z \ar[r]^-{\Delta _{Z/T}} \ar[r] & Z \times _ T Z }
is a fibre product diagram. Hence i is the base change of the diagonal morphism \Delta _{Z/T}. Thus the lemma follows from Lemma 67.4.1, and the material in Section 67.3.
\square
Comments (1)
Comment #1106 by Evan Warner on