changed the statement
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2018-05-05 |
a17c6b0 |
Remove math from \item[..]
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changed the statement
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2011-08-13 |
4ea0b65 |
Whitespace changes
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changed the statement and 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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2010-10-09 |
2b090dd |
End conversion of etale to \'etale.
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changed the statement and the proof
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2010-06-06 |
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 statement and the proof
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2010-01-31 |
9e88016 |
Conditions on algebraic spaces renamed.
OK, after this commit (which is basically without mathematical
content) we now have the following notions:
very reasonable: this is the old notion of "reasonable" and means
the space has a Zariski covering such that each piece has a
quasi-compact etale covering by a scheme.
reasonable: this is the old notion of "almost reasonable" and
means that for every affine U and etale morphism U --> X the
fibres are universally bounded.
decent: this means that every point is representable by a
monomorphism from the spectrum of a field and that moreover this
monomorphism is quasi-compact.
Each of these is a very weak notion of separation on the
algebraic space. We have also defined what it means for a
morphism to have those properties (in terms of "fibres"). The
goal of making this change now is to prevent confusion when we
start adding material later, because we think that
decent/reasonable spaces will play a more important role than
very reasonable spaces.
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changed the statement and the proof
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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 and the proof
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2010-01-08 |
17e595e |
Properties of Spaces: Fix Lemmas 03JX and 03KE
Forced by mistake in previous lemma.
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changed the statement and the proof
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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 03JX
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2009-11-08 |
65620d4
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Tags: New tags added
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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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created statement with label lemma-bounded-fibres in spaces-properties.tex
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2009-11-06 |
f2f2181 |
Properties of Spaces: Finite fibres
A lemma on finite fibres of affines mapping in an etale manner
into an algebraic space. Before adding this to the online
project we should discuss bounds for fibres of quasi-finite
morphisms, perhaps by introducing a class of morphisms
characterized by having universally bounded fibres.
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