The Stacks project

LaTeX: \Spec

	Introduced the macro

	\def\Spec{\mathop{\rm Spec}}

	and changed all the occurences of \text{Spec} into \Spec.

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).

Properties of Spaces: Fix Lemma 03KF

	This lemma says that if X --> Y is representable and Y has
	property \beta, \gamma, \delta, \epsilon or \eta, then also X
	has that property.

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.

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.