changed the statement
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2015-04-20 |
ce95ef6 |
Lift specializations along flat morphisms of spaces
when the target is decent...
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changed the label to lemma-decent-specialization
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2012-05-16 |
7e05565 |
Improve chapter on decent spaces
A collection of things: get rid of the very reasonable material.
This is possible because we can now prove everything for
reasoble spaces which was previously only proved for very
reasonable spaces.
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changed the statement
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2012-05-16 |
7e05565 |
Improve chapter on decent spaces
A collection of things: get rid of the very reasonable material.
This is possible because we can now prove everything for
reasoble spaces which was previously only proved for very
reasonable spaces.
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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 statement
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2010-10-09 |
2b090dd |
End conversion of etale to \'etale.
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changed the label to lemma-very-reasonable-specialization
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2010-01-31 |
1642b95 |
Terminology changes:
"reasonable" ---> "very reasonable"
"almost reasonable" ---> "reasonable"
David Rydh suggested this change since the notion of being (what
is now called) very reasonable is not a particularly good
notion. On the other hand the notion of being (what is now
called) reasonable behaves quite well in various situations, and
it seems hard to envision results that use the assumption of
being very reasonable that do not hold for reasonable spaces.
Still, currently there are still some results of this form, so
we need to keep the notion "very reasonable" around (of course
we will always keep it around for the sake of referencing, but
in the future we may delegate it to a forgotten corner).
TODO (soon): Introduce decent spaces. These will be
characterized by having property (gamma).
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changed the statement
|
2010-01-31 |
1642b95 |
Terminology changes:
"reasonable" ---> "very reasonable"
"almost reasonable" ---> "reasonable"
David Rydh suggested this change since the notion of being (what
is now called) very reasonable is not a particularly good
notion. On the other hand the notion of being (what is now
called) reasonable behaves quite well in various situations, and
it seems hard to envision results that use the assumption of
being very reasonable that do not hold for reasonable spaces.
Still, currently there are still some results of this form, so
we need to keep the notion "very reasonable" around (of course
we will always keep it around for the sake of referencing, but
in the future we may delegate it to a forgotten corner).
TODO (soon): Introduce decent spaces. These will be
characterized by having property (gamma).
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changed the statement and the proof
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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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assigned tag 03IL
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2009-10-25 |
2ad4800
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Tags: New tags added and two fixed
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changed the proof
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2009-10-25 |
851829c |
Properties of Spaces: Cleanup of results so far
We added a remark recalling some of the pertinent facts about
etale morphisms of schemes in the section on points of
reasonable algebraic spaces. Then we use this to shorten the
proofs of the lemmas in that section. We added a lemma saying
that a reasonable algebraic space covered by the spectrum of a
field is the spectrum of a field. Finally, we changed the lemma
stating that the topological space associated to a reasonable
space is sober into stating that it is Kolmogorov. The proof was
incorrect, and it will require considerably more work to prove
this.
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changed the proof
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2009-10-22 |
c378ebd |
Properties of Spaces: Reasonable => sober
It is a little bit rough here and there and some references need
to be added. Also some of the arguments are duplicated.
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created statement with label lemma-reasonable-specialization in spaces-properties.tex
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2009-10-22 |
2bd9d8e |
Properties of Spaces: Reasonable spaces have sober underlying |X|
We have not yet completely proved this but it looks like it is
going to work out. This commit has two FIXMES
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