The Stacks project

Lift specializations along flat morphisms of spaces

when the target is decent...

Finished cleanup decent-space.tex

	Last edit to this chapter for now.

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.

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.

Morphisms of Spaces: Valuative criterion + finite separable extensions

	In the existence part of the valuative criterion for algebraic
	spaces it is enough to take a finite separable extension of the
	fraction field of the dvr. It is sometimes also necessary. It
	seems that if the morphism is separated, then it shouldn't be
	necessary, but we'll return to that later.

Morphisms of Spaces: Valuative criterion universal closednedd

	Finally, we have the other direction. We still have to
	reformulate this later for morphisms which are, say,
	quasi-separated.

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.