The Stacks project

Finished cleanup decent-space.tex

	Last edit to this chapter for now.

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.

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.

Conditions on algebraic spaces renamed.

	OK, after this commit (which is basically without mathematical
	content) we now have the following notions:

	very reasonable: this is the old notion of "reasonable" and means
	the space has a Zariski covering such that each piece has a
	quasi-compact etale covering by a scheme.

	reasonable: this is the old notion of "almost reasonable" and
	means that for every affine U and etale morphism U --> X the
	fibres are universally bounded.

	decent: this means that every point is representable by a
	monomorphism from the spectrum of a field and that moreover this
	monomorphism is quasi-compact.

	Each of these is a very weak notion of separation on the
	algebraic space. We have also defined what it means for a
	morphism to have those properties (in terms of "fibres"). The
	goal of making this change now is to prevent confusion when we
	start adding material later, because we think that
	decent/reasonable spaces will play a more important role than
	very reasonable spaces.

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).

Morphisms of spaces: Valuative criterion universal closedness revisited

	modified:   spaces-morphisms.tex
	modified:   spaces-properties.tex

Morphisms of spaces: More on relative conditions

Morphisms of Spaces: Relative conditions

	Trying to understand the relative versions of the local
	conditions found in Properties of Spaces

	Todo:
		Fix currently unfinished discussion of the above
		Add lemma about algebraic spaces etale over fields
		When does an algebraic space satisfy the sheaf
			condition for fpqc-coverings? This is missing in
			the discussion of algebraic space in the
			introductory chapter on algebraic spaces, but it
			doesn't have a high priority.
		Add remark discussing informally the relative conditions
			and what to do with them.