The Stacks project

Strong formal effectiveness => openness versality

Actually not hard to prove and you only need to check on first order
thickenings.

Bhargav's example from c31011d shows that strong formal effectiveness
does not always hold. But what about strong formal effectiveness
where the thickenings are always first order (between any two, not
just between consecutive indices)? This is the only thing needed for
the argument here, so it would be nice if it was true for algebraic
stacks in general.

Tags: Added new tags

First version of Artin's theorems

	This monster commit contains the first version of Artin's
	theorems on representability of stacks and spaces. It is still a
	bit rough.

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.