The Stacks project

Fix proof of existence cohen ring for any field

OK, now it is correct but to me it feels like the proof is far too
complicated relying on too much theory. Oh well.

THanks to Mark
https://stacks.math.columbia.edu/tag/0328#comment-6049

Small changes

Algebra: constructing flat weakly unramified local ring maps

	Given a local ring R and a residue field extension k -> k' there
	exists a flat local ring map R -> R' such that the maximal ideal
	of R' is generated by the maximal ideal of R and such that the
	residue field extension is the given field extension

	Having this the proof of the existence of Cohen rings is
	clarified.

	modified:   algebra.tex

Etale: Started tying the chapter on etale morphisms into the project

	modified:   algebra.tex
	modified:   etale.tex

Added new tags

	modified:   tags/tags

More stuff in algebra.tex:

	Integral closure is transitive
	Surjectivity of completion of surjective maps?
	Definition of I-adically complete modules
	Criterion as to when completion is complete
	Completion complete in Noetherian case
	Finiteness criterion (still not done)
	Residue field extension of map dvrs bounded
	Complete local ring with finitely generated maximal ideal is
		Noetherian
	Lemmas on Japanese property
	Starting to prove Tate's theorem on Japaneseness of complete
		rings

	modified:   algebra.tex

Continuing adding material to algebra.tex:

	Power series ring over a Noetherian ring is Noetherian
	Integral closure commutes with localization
	Completions of Noetherian rings are Noetherian
	Taking modules of differentials commutes with colimits
	Better characterization of formally smooth ring maps
	Short exact sequence of modules differentials associated to
		formally smooth ring map
	Formal smoothness lifts along small deformations
	Bunch of lemmas with goal to characterize separable field
		extensions as formally smooth field extensions
	Cohen rings are formally smooth
	Proof of Cohen structure theorem

	modified:   algebra.tex
	modified:   spaces.tex

Starting to write more algebra:
	Transcendence degree of field extensions
	Separability of field extensions
	Modified section on base change of algebras over fields
	Formal smoothness for extensions of fields
	Cohen rings
	Cohen structure theorem
	Nagata rings
	Japanese rings
	Ascent of normality

	modified:   algebra.tex