The Stacks project

amalg and coprod

Whitespace changes

End conversion of etale to \'etale.

Start fixing etale localization

	Moved the Keel-Mori lemma to More on Groupoids in Spaces.

	Fixed the first (much easier) lemma on etale localization of
	groupoids quasi-finite over a base. The error was from not
	thinking straight about arrows in a groupoid category. This
	first lemma is used in the chapter on morphisms of algebraic
	spaces to prove that an algebraic space separated and locally
	quasi-finite over a scheme is a scheme. Hence this lemma cannot
	be moved to More on Groupoids of Spaces, because it is used
	earlier.

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.

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.

Tags: new tags added

	modified:   tags/tags

Groupoids: Etale localization of groupoids

	This is fun, since we actually have a good theory of etale
	localization of schemes, and it more or less automatically gives
	good results for etale localization of (quasi-finite) groupoids

	modified:   groupoids.tex