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The Stacks project

History of tag 03Y3

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type time link
changed the statement 2024-05-16 a20cecb
Insert missing s

Thanks to Simon Vortkamp
https://stacks.math.columbia.edu/tag/03XV#comment-8847
changed the statement and the proof 2021-05-02 297c5b6
Clarify what is needed in bootstrap theorem

Thanks to DatPham
https://stacks.math.columbia.edu/tag/03Y3#comment-6069
changed the statement 2011-08-11 f496b59
LaTeX: \Sch

	Introduced a new macro

	\def\Sch{\textit{Sch}}

	and replaced all the occurences of \textit{Sch} with \Sch.
changed the statement and the proof 2010-10-09 97a5c76
Begin translating etale to \'etale or \acute{e}tale (in Math mode).
changed the proof 2010-01-27 a74ca1c
More morphisms: Cohen-Macaulay morphisms

	Cohen-Macaulay morphisms are fppf local on the target and
	syntomic local on the source.
moved the statement to file bootstrap.tex 2010-01-25 79c5606
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.
changed the proof 2010-01-25 79c5606
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.
assigned tag 03Y3 2010-01-03 92b033f
Tags: added new tags
changed the proof 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.
created statement with label theorem-bootstrap in spaces-morphisms.tex 2009-12-21 b7c3159
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.