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

Fixing references

	This time we're fixing the type of the reference

End conversion of etale to \'etale.

Groupoids: Put advanced material on groupoids in separated chapter

	We will rewrite the technical lemmas, the slicing lemma, and
	etale localization lemmas in order to fix errors and for
	clarity.

Tags: added new tags

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?

Morphisms of Spaces: Mistake in previous commit

	Too enthusiastic last night...

Morphisms of Spaces: Bootstrap, first version

	Here we finally prove the long awaited fact that a sheaf for the
	fppf topology whose diagonal is representable by algebraic
	spaces, and which has an etale surjective covering by an
	algebraic space, is itself an algebraic space. We deduce this
	from the, itself interesting, proposition that an algebraic
	space which is separated and locally quasi-finite over a scheme
	is itself a scheme.

	Both of these results are essential for being able to continue
	to the next level, uh, I mean algebraic stacks.