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changed the proof 2020-11-16 db61e94
Improve lemma slightly

Thanks to Zhenhua Wu
https://stacks.math.columbia.edu/tag/03GN#comment-5540
changed the proof 2020-11-16 c3cb5e7
Missing \dim_k inserted 3x

Thanks to Michael
https://stacks.math.columbia.edu/tag/03J6#comment-5538
changed the proof 2011-08-10 65ce54f
LaTeX: \Spec

	Introduced the macro

	\def\Spec{\mathop{\rm Spec}}

	and changed all the occurences of \text{Spec} into \Spec.
changed the statement 2009-12-23 2f54d62
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.
assigned tag 03J6 2009-11-08 65620d4
Tags: New tags added
changed the statement 2009-11-08 e545e01
Properties of Spaces: Split out arguments on points of spaces

	The purpose of this commit is to work out in more detail the
	arguments that lead to the result that a reasonable algebraic
	space X has a sober space of points |X|.

	In this reworking we discover the notion of an ``almost
	reasonable space''. An algebraic space X is almost reasonable if
	for every affine scheme U and etale morphism U --> X the fibres
	of U --> X are universally bounded.

	Later we will encouter the following question: Suppose given a
	fibre square diagram

		X' --> X
		|      |
		v      V
		V' --> V

	with V' --> V a surjective etale morphism of affine schemes,
	such that X' is reasonable. Is X reasonable? If you know how to
	(dis)prove this then please email stacks.project@gmail.com

	Anyway, the corresponding result for ``almost reasonable''
	spaces is easy. Moreover, an almost reasonable space is a
	colimit of quasi-separated algebraic spaces.

	But on the other hand, we do not know how to prove that an
	almost reasonable space X has an open dense subspace which is a
	scheme, nor do we know how to prove that |X| is sober.
created statement with label lemma-composition-universally-bounded in morphisms.tex 2009-11-06 ee1bc94
Morphisms: Morphisms with universally bounded fibres

	Is there a more suitable name for this property? If so, please
	email stacks.project@gmail.com.