The Stacks project

Another variant of the valuative criterion

We could require the K's to be function fields of
irreducible components of U...

Not proper but closed; small change

Adding labels to equations

	Preparation for update python scripts

Morphisms: Image of a proper scheme is proper

	Unexplicably, this was missing sofar from the discussion of
	proper morphisms. We also use this to slightly simplify the
	proof of the characterization of proper morphisms in the chapter
	on limits of schemes.

	modified:   limits.tex
	modified:   morphisms.tex

Started tags infrastructure

	new file:   scripts/add_tags.py
	modified:   scripts/functions.py
	new file:   tags/initial_tags
	new file:   tags/tags

More neurotic changes

More neurotic changes

Finished tying loose ends relating Krull-Akizuki and valuative criterion
for proper morphisms over Noetherian bases

	modified:   algebra.tex
	modified:   constructions.tex
	modified:   limits.tex

Small changes to text added in previous commit and addition of a section
to the chapter on algebra regarding Krull-Akizuki

TODO in algebra.tex

	State and prove Krull-Akizuki
	Construct dvrs using blow-ups
	Move section on Krull-Akizuki if needed

	modified:   algebra.tex
	modified:   limits.tex

Proofs of three applications of CHow's lemma by Daniel Disengi, You Qi,
and Zachary Maddock

	modified:   limits.tex

Statement of applications of Chow's lemma

	modified:   limits.tex