The Stacks project

Remove math from \item[..]

Whitespace changes

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.

Etale local on the source-and-target

	Absurdly detailed discussion of three different notions of what
	it can mean for a property of morphisms of schemes to be etale
	local on the source and target. We choose the strongest of the
	three to avoid confusion that will inevitably arise when picking
	one of the other two. Moreover, it will be nicely compatible
	with the notion (to be introduced in the next commit) of what it
	means for a property of morphisms of germs to be etale local on
	the source and target.

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).

Properties of Spaces: Fix Lemmas 03JX and 03KE

	Forced by mistake in previous lemma.

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.

Properties of Spaces: Finite fibres

	A lemma on finite fibres of affines mapping in an etale manner
	into an algebraic space. Before adding this to the online
	project we should discuss bounds for fibres of quasi-finite
	morphisms, perhaps by introducing a class of morphisms
	characterized by having universally bounded fibres.