History of tag 040I
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changed the proof
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2013-05-12 |
b078e9c |
Fix a statement in topologies.tex
Also added a few of the omitted proofs
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assigned tag 040I
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2010-01-17 |
aeaa969
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Tags: added new tags
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created statement with label lemma-disjoint-union-is-fpqc-covering in topologies.tex
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2010-01-15 |
bb61741 |
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.
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