The Stacks project

Fun improvement characterization affine schemes

Thanks to Xuande Liu and Tongke Tang
https://stacks.math.columbia.edu/tag/05YU#comment-6678

Slogans by Brian Lawrence edited by Johan de Jong

Thanks to Brian Lawrence

The case of H^0 of Gabbers result

Ample on reduction is ample

X --> Y surjective integral and X affine, then Y affine

	for algebraic spaces. Finally!

Whitespace changes

LaTeX: \Spec

	Introduced the macro

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

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

LaTeX: fix colim

	Introduced the macro

	\def\colim{\mathop{\rm colim}\nolimits}

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

LaTeX: fix lim

	Replaced all the occurences of \text{lim} by \lim or
	\lim\nolimits depending on whether the invocation occured in
	display math or not.

Tags: Added new tags

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...