The Stacks project

Fix error in proof lemma

Thanks to 李一笑 who wrote

"In the proof of lemma 0241, one applies $\phi'$ to $(v_i,x_j)$, which
may not be a well defined $V' \times_{S'} X'$-point."

Unfortunately, the lemma wasn't fixable and we needed to change the
statement of the lemmma as well.

A counter example to the original statement of the lemma is for example
given by taking a nonempty scheme S and taking both X and S' to be two
disjoint copies of S mapping to S and finally taking both V and W to be
two copies of X mapping to X with the descent datum that produces
again two copies of S mapping to S when you descend V and W along X
mapping to S. Then you see that a morphism of descent data from V' to W'
just comes down to a morphism of two copies of S' over S' to two copies
of S' over S' which doesn't always descend to a corresponding morphism
over S... (This is just to remind me what went wrong; when I wrote this
I must have thought that the condition that X' x_{S'} X' ---> X x_S X
would prevent this kind of thing, but it was obviously garbage.)

Whitespace changes

Descent: morphisms of schemes satisfy fpqc descent

	We edited the descent chapter to improve our exposition,
	prompted by a question of Thanos D. Papaïoannou, namely

	"Do morphisms of algebraic spaces satisfy fpqc descent?"

	Here is one answer:

	Suppose X, Y are algebraic spaces over an affine base S. Suppose
	that S' --> S is a surjective flat morphism of affine schemes.
	Finally suppose that a' : X_{S'} --> Y_{S'} is a morphism of
	algebraic spaces over S' which is compatible with the canonical
	descent data. Now you want to know if a' descends to a morphism
	a : X --> Y over S.

	If X, Y are representable, then this is Descent, Lemma Tag 02W0.
	See also the explanation in the remark following that lemma. The
	proof is kind of long since I tried to prove somehow more
	general statements in that section. The key is Descent Lemma Tag
	0241. It mainly relies on the fact that the representable
	presheaf associated to a scheme satisfies the sheaf condition
	for the fpqc topology (which is Descent, Lemma Tag 023Q). The
	analogue for this in the category of algebraic spaces is
	Properties of Spaces, Lemma Tag 03WB but there is a condition,
	namely that the algebraic space is Zariski locally quasi-separated.

	So I do not know how to prove this descent in general, and it
	may not be true. But it is true for Zariski locally
	quasi-separated spaces. Eventually we will state this descent
	property explicitly in the stacks project.

	Any algebraic space in the literature, say published before year
	2000, is quasi-separated (since there are basically no
	references which deal with more general ones). Hence fpqc
	descent for morphisms of algebraic spaces is true for any
	algebraic spaces which you can find in these articles/books.

Renamed fpqc-descent.tex descent.tex

	modified:   Makefile
	modified:   algebraic.tex
	modified:   browse.html
	modified:   chapters.tex
	renamed:    fpqc-descent.tex -> descent.tex
	modified:   groupoids.tex
	modified:   more-morphisms.tex
	modified:   morphisms.tex
	modified:   preamble.tex
	modified:   scripts/functions.py
	modified:   spaces.tex
	modified:   stacks.tex
	modified:   tags/Makefile
	modified:   tags/tags

Got to the point where separated etale morphisms satsify descent for
fppf coverings (not fpqc coverings though)

	modified:   algebra.tex
	modified:   fpqc-descent.tex
	modified:   more-morphisms.tex
	modified:   morphisms.tex

Descending properties of morphisms and related lemmas

	modified:   algebra.tex
	modified:   fpqc-descent.tex
	modified:   morphisms.tex

Started tags infrastructure

	new file:   scripts/add_tags.py
	modified:   scripts/functions.py
	new file:   tags/initial_tags
	new file:   tags/tags

More neurotic changes

Moved material on descending schemes into fpqc-descent.tex

	modified:   fpqc-descent.tex