The Stacks project

Reference for 04DZ.

Second proof for topological invariance etale site

This is the proof of the equivalence of etale sites under a
universal homeomorphism. The first proof uses the very delicate
arguments about comparison of sites in various forms earlier
in the chapter on etale cohomology. This material is kinda
hard to read and maybe a bit nonstandard. The new proof
is just like the proof in SGA 1 and uses the descent for
etale morphisms along surjective integral morphisms + the
topological invariance for thickenings... In fact in some
sense the new proof may very well be harder, but it seems
at least easier to break it up into parts which have meaning...

Start explaining Gabber's argument

pell check: words starting with n, o, p, q, r, N, O, P, Q, or R

Moving lemmas for clarity

	Following a suggestion of David Rydh we tried to collect results
	related to universally injective unramified morphisms into one
	place. We did not completely succeed, but hopefully the end
	result is still an improvement!

Wrap up proof

	of topological invariance of X_{spaces, etale} for integral,
	universally injective, and surjective morphisms. It seems to me
	this is an open question when you only assume the morphism is a
	universal homeomorphism... Anybody?

Bunch of changes

	(1) Starting to write about thickenings of algebraic spaces
	(2) Change Omega^1_{X/S} to Omega_{X/S}
	(3) Introduced universal homeomorphisms for algebraic spaces
	(4) If X ---> Y is a surjective, integral morphism of schemes
	and X is an affine scheme, then Y is an affine scheme
	(4) Topological invariance of the site X_{spaces, etale} of an
	algebraic space X (proof unfinished}

	In order to see that also X_{etale} is a topological invariant
	we (I think) need to prove the following result: If X --> Y is a
	integral, universally injective, surjective morphism of
	algebraic spaces then X is a scheme if and only if Y is a
	scheme. There are two proofs of this result in the literature
	(one by David Rydh and one by Brian Conrad); both reduce the
	result to the Noetherian case by limit arguments. I would
	prefer a more direct argument...

Begin translating etale to \'etale or \acute{e}tale (in Math mode).

Etale local on the source-and-target

	Absurdly detailed discussion of three different notions of what
	it can mean for a property of morphisms of schemes to be etale
	local on the source and target. We choose the strongest of the
	three to avoid confusion that will inevitably arise when picking
	one of the other two. Moreover, it will be nicely compatible
	with the notion (to be introduced in the next commit) of what it
	means for a property of morphisms of germs to be etale local on
	the source and target.

Exclamation --> period.

Large commit

	Just trying to recover from the brain dead mistakes made over
	the last few weeks. Hopefully most of it is now more or less OK.

Large commit

	Just trying to recover from the brain dead mistakes made over
	the last few weeks. Hopefully most of it is now more or less OK.