The Stacks project

Fix a few typos

Pointed out by Rex and David Holmes the comments (too many to list
here).

LaTeX: \Spec

	Introduced the macro

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

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

Big set of changes

	Most of this is geared towards proving (in the "correct" way)
	that an algebraic stack is a gerbe if and only if the inertia is
	flat and locally of finite presentation over it.

	Other additions are:
		stackification and inertia
		[U/G] ---> U/G in the case of free actions
		residual gerbes are gerbes (not nec over a point!)
		presentations of inertia stacks
		definition free action
		U ---> U/R is flat + lfp in final bootstrap theorem

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.

Added new tags to the project

	modified:   tags/tags

Some examples of algebraic spaces

	modified:   my.bib
	modified:   spaces.tex