The Stacks project

Improve chapter on decent spaces

	A collection of things: get rid of the very reasonable material.
	This is possible because we can now prove everything for
	reasoble spaces which was previously only proved for very
	reasonable spaces.

Cleanup in Decent Spaces

	More streamlined. We also (finally) made it precise that a space
	is decent if and only if every one of its points is given by a
	quasi-compact monomorphism from the spectrum of a field. We can
	probably use this fact to our advantage in a bunch of the proofs
	of this chapter...

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.

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.

Tags: New tags added

Properties of Spaces: Split out arguments on points of spaces

	The purpose of this commit is to work out in more detail the
	arguments that lead to the result that a reasonable algebraic
	space X has a sober space of points |X|.

	In this reworking we discover the notion of an ``almost
	reasonable space''. An algebraic space X is almost reasonable if
	for every affine scheme U and etale morphism U --> X the fibres
	of U --> X are universally bounded.

	Later we will encouter the following question: Suppose given a
	fibre square diagram

		X' --> X
		|      |
		v      V
		V' --> V

	with V' --> V a surjective etale morphism of affine schemes,
	such that X' is reasonable. Is X reasonable? If you know how to
	(dis)prove this then please email stacks.project@gmail.com

	Anyway, the corresponding result for ``almost reasonable''
	spaces is easy. Moreover, an almost reasonable space is a
	colimit of quasi-separated algebraic spaces.

	But on the other hand, we do not know how to prove that an
	almost reasonable space X has an open dense subspace which is a
	scheme, nor do we know how to prove that |X| is sober.