The Stacks project

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Getting smooth morphisms towards X

	Openness of versality gives this to us. But you have to worry a
	bit about (purely inseparable) residue field extensions, i.e.,
	you have to show that the versality condition is preserved under
	a bunch of reasonable operations.

	Somehow Artin never seems to run into this problem. I think
	because he consistently works with covers having a fixed residue
	field. So yes, it is possible to avoid this...

Formulating the axioms

	I've decided to formulate the axiom on effectivity as asking for
	an equivalence

		X(R) = lim X(R/m^n).

	There are two reasons for this: (1) this is what you get when X
	is an algebraic stack and (2) I don't know any case where the
	natural proof of the density that Artin requires, doesn't give
	you the equality as stated above. If you do know an example of
	this, please let me know.

	What I can imagine being easier to prove (in examples) is that
	versal deformations can be approximated over complete local
	rings. (For example, these complete local rings may have
	additional properties which help approximate the given formal
	objects.) In other words, it may make sense to have a second
	version of the theorem where we assume the existence of
	*effective* versal formal objects without assuming RS. And this
	also makes sense as after all Artin's approximation method takes
	as input such things...

	To be continued.

An example showing the G-ring hypothesis is necessary