The Stacks project

Codimension 0 points on algebraic spaces

This is a rather large set of changes all related to different ways of
saying what it means to have a ``generic point'' on an algebraic space.
One particularly nice lemma says that an algebraic space which locally
(in a suitable sense) has finitely many generic points is automatically
a reasonable algebraic space (in particular decent). The proof is the
same as the argument showing that a decent locally Noetherian algebraic
space is quasi-separated.

These types of results may be useful in the future as tools to decide
whether a given algebraic space is decent and/or quasi-separated.

Update a remark to incorporate more recent results

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.

LaTeX: \Spec

	Introduced the macro

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

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

Fix ordering of material in decent-spaces.tex

	This also means we're now down with the basic reorganization of
	the material in algebraic spaces. What is a bit unsatisfactory
	is that some basic material on lifting specializations is only
	done in the chapter on decent spaces and hence cannot be used
	(even for quasi-separated algebraic spaces) until after this
	chapter.

	Especially, the lemma on lifting specializations from an
	algebraic space to an etale cover should be formulated and
	proved for quasi-separated spaces (it should be as short a proof
	as possible).

Fixes in spaces-properties.tex

	We tried to find arguments for some of the previous results in
	the quasi-separated case which are easier than the original more
	general ones. We only partially succeeded. But on the other
	hand, we can in the future keep simplifying this chapter and add
	the more involved arguments to the chapter on decent spaces.