The Stacks project

Remove 'f.f.'

Sad IMHO.

Thanks to BCnrd, Dario Weissmann, and sdf
https://stacks.math.columbia.edu/tag/02JQ#comment-2762
https://stacks.math.columbia.edu/tag/02JQ#comment-2765
https://stacks.math.columbia.edu/tag/02JQ#comment-2766
https://stacks.math.columbia.edu/tag/02JQ#comment-2767

Tags: Added new tags

More results on formal smoothness

	In particular the characterization of formally smooth maps
	between Noetherian local rings in terms of flatness and fs
	fibre. The proof is a bit long (about 2 pages) but it isn't
	clear how to split it further without introducing substantially
	more notation.

Noetherian complete local rings are G-rings

	Finish the proof. This is IMHO somewhat easier to understand
	than the argument in Matsumura's book, as it uses the
	construction of derivations from an earlier section.

Noetherian complete local rings are G-rings

	Finish the proof. This is IMHO somewhat easier to understand
	than the argument in Matsumura's book, as it uses the
	construction of derivations from an earlier section.

Towards results on G-rings

	This material is surprisingly annoying to grok. For example, the
	correct way to proceed is undoubtedly to use Nagata's Jacobian
	criterion to show that rings of finite type over Noetherian
	complete local rings are G-rings. However, there seems to be no
	easy way to actually prove that the criterion applies...

	The algebra question that one gets is the following: Suppose
	that P is a prime ideal of height c in a ring of the form

		k[[x_1, ..., x_n]][y_1, ..., y_m]

	where k is either a field or a Cohen ring. Then we need to prove
	there are derivations D_1, ..., D_c of this ring such that the
	matrix

		D_i(f_j) mod P

	has rank c for some f_1, ..., f_c in P. Let me know if there is
	a simple proof of this result (currently I am not even 100% sure
	it is true).