The Stacks project

Algebraic spaces are always over a base scheme

Thanks to Kestutis Cesnavicius
http://stacks.math.columbia.edu/tag/03XX#comment-688

pell check: words starting with n, o, p, q, r, N, O, P, Q, or R

Small changes

Whitespace changes

Moving lemmas for clarity

	Following a suggestion of David Rydh we tried to collect results
	related to universally injective unramified morphisms into one
	place. We did not completely succeed, but hopefully the end
	result is still an improvement!

unramfied -> unramified

End conversion of etale to \'etale.

Global: Eradicated radicial

	Well, not completely, but almost.

Tags: added new tags

Morphisms of Spaces: Bootstrap, second version

	OK, so now the proof is complete. Of course the chapter on
	morphisms on algebraic spaces has a curious selection of topics
	at the moment, since we've tried to work towards the bootstrap
	theorem, and have not developped in a straightforward way. For
	example, we have at this point defined what an etale morphism of
	algebraic spaces is, but not what a morphism of finite
	presentation is!

	This will be fixed over time.

Morphisms of Spaces: Repaired lemma

	OK, so the lemma is fixed, and the proof is even kind of fun.
	The statement is that given a diagram

		V' ---> X' ---> X
			|	|
			v	v
			T' ---> T

	where T' --> T is an etale morphism of affine schemes, X an
	algebraic space, and X --> T is a separated locally quasi-finite
	morphism, and V' is an open subspace of X' which is a scheme and
	quasi-affine over T', then the image of V' in X is a scheme
	also. Of course this is subsumed in the final proposition that X
	in the situation above is actually a scheme!

	No matter: The question is whether one could prove a result like
	the above with a weakened hypothesis on the morphism X --> T.
	For example, suppose in the diagram above X --> T is only
	assumed quasi-separated, and we assume T' --> T is a finite
	etale Galois covering with group G. Then we can consider

		W' = \bigcap_{g \in G} g(V')

	This would still be quasi-affine over T', and its image in X a
	scheme, because W' comes with a natural descent datum for the
	morphism T' --> T.

	I assume that the condition of being Galois is too strong and it
	would suffice for T' --> T to just be finite locally free? But
	is there some part of this argument that survives if T' --> T is
	``just'' etale? Or is there some easy counter example?