History of tag 03JS
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type |
time |
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changed the proof
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2011-08-13 |
4ea0b65 |
Whitespace changes
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changed the proof
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2011-06-11 |
5619b77 |
Cleanup in Decent Spaces
More streamlined. We also (finally) made it precise that a space
is decent if and only if every one of its points is given by a
quasi-compact monomorphism from the spectrum of a field. We can
probably use this fact to our advantage in a bunch of the proofs
of this chapter...
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moved the statement to file decent-spaces.tex
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2011-06-10 |
52c6ad3 |
Decent Algebraic Spaces
Created a new chapter "Decent Algebraic Spaces" and moved most
of the material on local conditions of algebraic spaces in
there. In the next few commits we will fix the breakage that this
causes.
The reason for the move is that this material is difficult to
understand for the beginner and that most of the other material
in Properties of Spaces and Morphisms of Spaces is easier and
more analogous to what happens for schemes.
An added advantage is that we can use results on morphisms of
algebraic spaces in the new chapter, hence it becomes easier to
develop the theory of decent spaces.
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changed the proof
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2011-06-10 |
52c6ad3 |
Decent Algebraic Spaces
Created a new chapter "Decent Algebraic Spaces" and moved most
of the material on local conditions of algebraic spaces in
there. In the next few commits we will fix the breakage that this
causes.
The reason for the move is that this material is difficult to
understand for the beginner and that most of the other material
in Properties of Spaces and Morphisms of Spaces is easier and
more analogous to what happens for schemes.
An added advantage is that we can use results on morphisms of
algebraic spaces in the new chapter, hence it becomes easier to
develop the theory of decent spaces.
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changed the statement and the proof
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2010-10-09 |
2b090dd |
End conversion of etale to \'etale.
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changed the statement and the proof
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2010-01-08 |
617b8ab |
Properties of Spaces: Fix error in lemma
David Rydh pointed out that the lemma fixed in this commit was
wrong and suggested the correction.
This mistake has some ramifications for later results which we
will correct in the next few commits.
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assigned tag 03JS
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2009-11-08 |
65620d4
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Tags: New tags added
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created statement with label lemma-U-finite-above-x in spaces-properties.tex
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2009-11-08 |
e545e01 |
Properties of Spaces: Split out arguments on points of spaces
The purpose of this commit is to work out in more detail the
arguments that lead to the result that a reasonable algebraic
space X has a sober space of points |X|.
In this reworking we discover the notion of an ``almost
reasonable space''. An algebraic space X is almost reasonable if
for every affine scheme U and etale morphism U --> X the fibres
of U --> X are universally bounded.
Later we will encouter the following question: Suppose given a
fibre square diagram
X' --> X
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v V
V' --> V
with V' --> V a surjective etale morphism of affine schemes,
such that X' is reasonable. Is X reasonable? If you know how to
(dis)prove this then please email stacks.project@gmail.com
Anyway, the corresponding result for ``almost reasonable''
spaces is easy. Moreover, an almost reasonable space is a
colimit of quasi-separated algebraic spaces.
But on the other hand, we do not know how to prove that an
almost reasonable space X has an open dense subspace which is a
scheme, nor do we know how to prove that |X| is sober.
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