The Stacks project

fix small typos

Add a small lemma to spaces-properties

Thanks to Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/03E1#comment-7758

Another characterization of quasi-compact points

Thanks to Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/03JV#comment-7747

Another characterization of decent points

For some reason this was missing...

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.

Properties of Spaces: Added another condition to Lemma 03JS

	As mentioned in commit 2e77361c58f0a32c168a8e5fdc6c67b91d5038cd

Properties of Spaces: Fix next lemma.

	Lemma 03JV was wrong since it used the previously erroneous
	Lemma 03JS. We split of the weaker condition Lemma
	\ref{lemma-weak-UR-finite-above-x} (does not have a tag yet)
	and corrected Lemma 03JS by removing two of the four conditions.

	Presumably we can add another condition to the list of
	equivalent conditions of Lemma 03JS, namely as suggested by
	David Rydh:

	"there exists a family of etale morphisms U_i --> X which is
	jointly surjective and such that for each i the fibres of
		U_i --> X  and  U_i \times_X U_i --> X
	over x are finite (as in Lemma 03JT and the fixed Lemma 03JS)"

	This should go on the todo list.

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.

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.