History of tag 03WT
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changed the statement
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2010-10-09 |
2b090dd |
End conversion of etale to \'etale.
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assigned tag 03WT
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2010-01-03 |
92b033f
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Tags: added new tags
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created statement with label lemma-etale-open in morphisms.tex
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2009-12-22 |
1c615f0 |
Morphisms of Spaces: Repaired lemma
OK, so the lemma is fixed, and the proof is even kind of fun.
The statement is that given a diagram
V' ---> X' ---> X
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v v
T' ---> T
where T' --> T is an etale morphism of affine schemes, X an
algebraic space, and X --> T is a separated locally quasi-finite
morphism, and V' is an open subspace of X' which is a scheme and
quasi-affine over T', then the image of V' in X is a scheme
also. Of course this is subsumed in the final proposition that X
in the situation above is actually a scheme!
No matter: The question is whether one could prove a result like
the above with a weakened hypothesis on the morphism X --> T.
For example, suppose in the diagram above X --> T is only
assumed quasi-separated, and we assume T' --> T is a finite
etale Galois covering with group G. Then we can consider
W' = \bigcap_{g \in G} g(V')
This would still be quasi-affine over T', and its image in X a
scheme, because W' comes with a natural descent datum for the
morphism T' --> T.
I assume that the condition of being Galois is too strong and it
would suffice for T' --> T to just be finite locally free? But
is there some part of this argument that survives if T' --> T is
``just'' etale? Or is there some easy counter example?
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