changed the statement and the proof
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2021-05-02 |
7d60276 |
Clarify what the argument actually shows
Thanks to comment_bot
https://stacks.math.columbia.edu/tag/03K0#comment-6119
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changed the proof
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2011-08-11 |
f496b59 |
LaTeX: \Sch
Introduced a new macro
\def\Sch{\textit{Sch}}
and replaced all the occurences of \textit{Sch} with \Sch.
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changed the proof
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2011-08-10 |
65ce54f |
LaTeX: \Spec
Introduced the macro
\def\Spec{\mathop{\rm Spec}}
and changed all the occurences of \text{Spec} into \Spec.
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changed the statement and the proof
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2011-08-10 |
996a95d |
LaTeX: fix colim
Introduced the macro
\def\colim{\mathop{\rm colim}\nolimits}
and changed all the occurences of \text{colim} into \colim.
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changed the proof
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2011-06-17 |
1692461 |
Neurotic changes
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changed the label to lemma-fun-property-reasonable
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2011-06-12 |
b88197a |
Finished cleanup decent-space.tex
Last edit to this chapter for now.
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changed the statement and the proof
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2011-06-12 |
b88197a |
Finished cleanup decent-space.tex
Last edit to this chapter for now.
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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 statement
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2010-06-06 |
b504204 |
Etale local on the source-and-target
Absurdly detailed discussion of three different notions of what
it can mean for a property of morphisms of schemes to be etale
local on the source and target. We choose the strongest of the
three to avoid confusion that will inevitably arise when picking
one of the other two. Moreover, it will be nicely compatible
with the notion (to be introduced in the next commit) of what it
means for a property of morphisms of germs to be etale local on
the source and target.
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changed the label to remark-fun-property-reasonable
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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
|
2009-11-09 |
15b591e |
Morphisms of Spaces: Valuative criterion universal closednedd
Finally, we have the other direction. We still have to
reformulate this later for morphisms which are, say,
quasi-separated.
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assigned tag 03K0
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2009-11-08 |
65620d4
|
Tags: New tags added
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created statement with label remark-fun-property-almost-reasonable 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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