The Stacks project

Insert missing s

Thanks to Simon Vortkamp
https://stacks.math.columbia.edu/tag/03XV#comment-8847

Clarify what is needed in bootstrap theorem

Thanks to DatPham
https://stacks.math.columbia.edu/tag/03Y3#comment-6069

LaTeX: \Sch

	Introduced a new macro

	\def\Sch{\textit{Sch}}

	and replaced all the occurences of \textit{Sch} with \Sch.

Begin translating etale to \'etale or \acute{e}tale (in Math mode).

More morphisms: Cohen-Macaulay morphisms

	Cohen-Macaulay morphisms are fppf local on the target and
	syntomic local on the source.

Bootstrap: Added new chapter

	It seems better to discuss the bootstrap theorems in a separate
	chapter following the discussion of the basic material on
	algebraic spaces. This can then be used in the chapter on
	algebraic stacks when discussing presentations and what not.

	The goal is to prove the theorem that an fppf sheaf F for which
	there exists a scheme X and a morphism f : X --> F such that
		f is representable by algebraic spaces
		f is flat, surjective and locally of finite presentation
	is automatically an algebraic space. We have almost all the
	ingredients ready except for a slicing lemma -- namely somehow
	lemma 3.3 part (1) of Keel-Mori.

Bootstrap: Added new chapter

	It seems better to discuss the bootstrap theorems in a separate
	chapter following the discussion of the basic material on
	algebraic spaces. This can then be used in the chapter on
	algebraic stacks when discussing presentations and what not.

	The goal is to prove the theorem that an fppf sheaf F for which
	there exists a scheme X and a morphism f : X --> F such that
		f is representable by algebraic spaces
		f is flat, surjective and locally of finite presentation
	is automatically an algebraic space. We have almost all the
	ingredients ready except for a slicing lemma -- namely somehow
	lemma 3.3 part (1) of Keel-Mori.

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: 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.