The Stacks project

Add missing symbol in spaces-properties

Codimension 0 points on algebraic spaces

This is a rather large set of changes all related to different ways of
saying what it means to have a ``generic point'' on an algebraic space.
One particularly nice lemma says that an algebraic space which locally
(in a suitable sense) has finitely many generic points is automatically
a reasonable algebraic space (in particular decent). The proof is the
same as the argument showing that a decent locally Noetherian algebraic
space is quasi-separated.

These types of results may be useful in the future as tools to decide
whether a given algebraic space is decent and/or quasi-separated.

Quotients which are schemes

	Just a clear statement of what we've already proved.

Neurotic changes

Tags: Added new tags

Fixes in spaces-properties.tex

	We tried to find arguments for some of the previous results in
	the quasi-separated case which are easier than the original more
	general ones. We only partially succeeded. But on the other
	hand, we can in the future keep simplifying this chapter and add
	the more involved arguments to the chapter on decent spaces.

Fixes in spaces-properties.tex

	We tried to find arguments for some of the previous results in
	the quasi-separated case which are easier than the original more
	general ones. We only partially succeeded. But on the other
	hand, we can in the future keep simplifying this chapter and add
	the more involved arguments to the chapter on decent spaces.

Decent Algebraic Spaces

	Created a new chapter "Decent Algebraic Spaces" and moved most
	of the material on local conditions of algebraic spaces in
	there. In the next few commits we will fix the breakage that this
	causes.

	The reason for the move is that this material is difficult to
	understand for the beginner and that most of the other material
	in Properties of Spaces and Morphisms of Spaces is easier and
	more analogous to what happens for schemes.

	An added advantage is that we can use results on morphisms of
	algebraic spaces in the new chapter, hence it becomes easier to
	develop the theory of decent spaces.

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.

Terminology changes:
	"reasonable" ---> "very reasonable"
	"almost reasonable" ---> "reasonable"

	David Rydh suggested this change since the notion of being (what
	is now called) very reasonable is not a particularly good
	notion. On the other hand the notion of being (what is now
	called) reasonable behaves quite well in various situations, and
	it seems hard to envision results that use the assumption of
	being very reasonable that do not hold for reasonable spaces.

	Still, currently there are still some results of this form, so
	we need to keep the notion "very reasonable" around (of course
	we will always keep it around for the sake of referencing, but
	in the future we may delegate it to a forgotten corner).

	TODO (soon): Introduce decent spaces. These will be
	characterized by having property (gamma).

Terminology changes:
	"reasonable" ---> "very reasonable"
	"almost reasonable" ---> "reasonable"

	David Rydh suggested this change since the notion of being (what
	is now called) very reasonable is not a particularly good
	notion. On the other hand the notion of being (what is now
	called) reasonable behaves quite well in various situations, and
	it seems hard to envision results that use the assumption of
	being very reasonable that do not hold for reasonable spaces.

	Still, currently there are still some results of this form, so
	we need to keep the notion "very reasonable" around (of course
	we will always keep it around for the sake of referencing, but
	in the future we may delegate it to a forgotten corner).

	TODO (soon): Introduce decent spaces. These will be
	characterized by having property (gamma).

Properties of Spaces: Improve statement of claim in Proposition 03JI

	CLAIM: Suppose you have a groupoid scheme (U, R, s, t,...) which
	defines an etale equivalence relation with s, t finite (!). Then
	the union of the R-invariant affine opens of U is topologically
	dense in U.

	Here is a discussion of the proof which may be clearer than what
	is now actually in the stacks project:

	If U is separated this is easy because a finite set of generic
	points of U is contained in an affine open, and generic points
	are dense. Hence given a generic point of U you take an affine W
	containing its orbit $O$ (which is a finite set of generic
	points of U), and then you find an invariant affine inside W
	containing this orbit $O$ by standard arguments. The point (in
	the proof of the Proposition) is that such an argument also
	works if U is arbitrary by the silly trick which says that any
	quasi-compact scheme has a dense open which is separated...

Tags: New tags added

Properties of Spaces: Reasonable spaces have dense schematic locus

	This is kind of tricky...