The Stacks project

Ample on reduction is ample

LaTeX: \Spec

	Introduced the macro

	\def\Spec{\mathop{\rm Spec}}

	and changed all the occurences of \text{Spec} into \Spec.

More fixes of short titles

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!

Tags: Added new tags

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...