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.

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.

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).

Tags: New tags added and two fixed

Properties of Spaces

	We started to work out the suggestion in commit 2100745. In fact
	the suggestion was wrong and the correct notion is to require
	that there exists a surjective etale morphism \coprod U_i --> X
	such that for each i the two projection morphisms
		U_i \times_X U_i --> U_i
	are quasi-compact. We are calling such an algebraic space
	``reasonable''. If you do not like this please complain soon.
	Sofar the only interesting observation is that points on
	reasonable spaces are represented by monomorphisms from spectra
	of fields. We also expect that valuative criteria will work well
	for reasonable algebraic spaces.