The Stacks project

ell check: words starting with s, t, S, or T

LaTeX: \Spec

	Introduced the macro

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

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

Finished cleanup decent-space.tex

	Last edit to this chapter for now.

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).

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.

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.

Tags: Added new tags

	Also commented obsolete tags corresponding to sections of
	desirables.tex that have been removed.

Morphisms of spaces: Valuative criterion universal closedness revisited

	modified:   spaces-morphisms.tex
	modified:   spaces-properties.tex