The Stacks project

Add a small lemma to spaces-properties

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

Clarify statement lemma in spaces-properties

Clarify statement lemma in spaces-properties

Solve the problem of having duplicate tags

We could consider not adding the obsolete chapter to the book version.

LaTeX: \Spec

	Introduced the macro

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

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

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.

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.

Properties of Spaces: Cleanup of results so far

	We added a remark recalling some of the pertinent facts about
	etale morphisms of schemes in the section on points of
	reasonable algebraic spaces. Then we use this to shorten the
	proofs of the lemmas in that section. We added a lemma saying
	that a reasonable algebraic space covered by the spectrum of a
	field is the spectrum of a field. Finally, we changed the lemma
	stating that the topological space associated to a reasonable
	space is sober into stating that it is Kolmogorov. The proof was
	incorrect, and it will require considerably more work to prove
	this.

Properties of Spaces: Reorganization of material

	Discuss points on reasonable spaces in its own section.

Properties of Spaces: Reasonable spaces have sober underlying |X|

	We have not yet completely proved this but it looks like it is
	going to work out. This commit has two FIXMES

Properties of Spaces

	We started to work out the suggestion in commit 2100745. In fact
	the suggestion was wrong and the correct notion is to require
	that there exists a surjective etale morphism \coprod U_i --> X
	such that for each i the two projection morphisms
		U_i \times_X U_i --> U_i
	are quasi-compact. We are calling such an algebraic space
	``reasonable''. If you do not like this please complain soon.
	Sofar the only interesting observation is that points on
	reasonable spaces are represented by monomorphisms from spectra
	of fields. We also expect that valuative criteria will work well
	for reasonable algebraic spaces.

TAGS: added new tags

	modified:   tags/tags

Properties of spaces: Add some simple definitions

	modified:   spaces-properties.tex

Properties of spaces: Points of quasi-separated spaces behave well

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