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Ilya Zakharevich
science forum beginner
Joined: 20 May 2005
Posts: 27
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 Posted: Fri May 20, 2005 9:20 am Post subject:
Open but not universally open?
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"Any decent" book on algebraic geometry contains a statement that
"universally open mapping" (i.e., a mapping such that any base change
is an open mapping) is a strictly stronger condition than "open
mapping". However, it looks like the examples got lost in the
darkness of centuries; I could not find any valid written example of
an open mapping which is not universally open.
Googling finds one paper which contains such an "example", but after
some discussion the author agreed that the example is not valid, and
said that he can't recall any example. (I mean
A_straight_way_to_algebraic_stacks.)
So: what are examples of such mappings? If possible, provide some
references too (especially if I missed some "obvious places").
Thanks,
Ilya |
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Jannick Asmus
science forum Guru
Joined: 25 Mar 2005
Posts: 312
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| Posted: Sun May 22, 2005 11:28 pm Post subject:
Re: Open but not universally open?
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On 20.05.2005 13:20, Ilya Zakharevich wrote:
| Quote: | "Any decent" book on algebraic geometry contains a statement that
"universally open mapping" (i.e., a mapping such that any base change
is an open mapping) is a strictly stronger condition than "open
mapping". However, it looks like the examples got lost in the
darkness of centuries; I could not find any valid written example of
an open mapping which is not universally open.
Googling finds one paper which contains such an "example", but after
some discussion the author agreed that the example is not valid, and
said that he can't recall any example. (I mean
A_straight_way_to_algebraic_stacks.)
So: what are examples of such mappings? If possible, provide some
references too (especially if I missed some "obvious places").
Thanks,
Ilya
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Who else could provide an example: Of course, it is A. Grothendieck.
Have a look at EGA IV.3, Remarque 14.3.9.i
(http://www.numdam.org/item?id=PMIHES_1966__28__5_0).
Cheers,
J. |
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Ilya Zakharevich
science forum beginner
Joined: 20 May 2005
Posts: 27
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| Posted: Tue May 24, 2005 2:30 pm Post subject:
Re: Open but not universally open?
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[A complimentary Cc of this posting was sent to
Jannick Asmus
<jannick.news@web.de>], who wrote in article <d6rbj6$84r$1@dizzy.math.ohio-state.edu>:
| Quote: | darkness of centuries; I could not find any valid written example of
an open mapping which is not universally open.
Who else could provide an example: Of course, it is A. Grothendieck.
Have a look at EGA IV.3, Remarque 14.3.9.i
(http://www.numdam.org/item?id=PMIHES_1966__28__5_0).
|
The example turns out to be quite simple (it was in the zoo of
non-flat mappings I have, but I did not realized it is not univerally
open): take a simplest example of non-flat mapping of curves:
normalization of a double point N --> X. By trivial reasons (it sends
finite sets to finite sets) it is an open mapping. Pulling back to N,
one gets a mapping N' --> N; over two "special" point of N it has two
preimages, over the complement it is a local isomorphism.
It is easy to calculate that N' is a disjoint union of N and two more
point; these points project to the "special" point of N. Obviously,
the mapping N' --> N is not open.
[My first impression is that one does not even need to work in
algebraic geometry settings to get this example; if one considers N
and X with Zarisky topology, it looks like the pullback in category
Top (of topological spaces) is also not open.
Note that the forgetting functor from category of schemes to one of
topological spaces is not left (?) exact; this inverse limits taken
in these categories are in general not-compatible. For example, the
Zarisky topology on the scheme A2 = A1 x A1 does not coincide with
direct square of a topological space A1 with Zariski topology.]
Thanks,
Ilya |
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Ilya Zakharevich
science forum beginner
Joined: 20 May 2005
Posts: 27
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| Posted: Tue May 24, 2005 8:00 pm Post subject:
Re: Open but not universally open?
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I wrote in article <d6vdpc$oai$1@news.ks.uiuc.edu>:
| Quote: | The example turns out to be quite simple (it was in the zoo of
non-flat mappings I have, but I did not realized it is not univerally
open): take a simplest example of non-flat mapping of curves:
normalization of a double point N --> X. By trivial reasons (it sends
finite sets to finite sets) it is an open mapping. Pulling back to N,
one gets a mapping N' --> N; over two "special" point of N it has two
preimages, over the complement it is a local isomorphism.
It is easy to calculate that N' is a disjoint union of N and two more
point; these points project to the "special" point of N. Obviously,
the mapping N' --> N is not open.
[My first impression is that one does not even need to work in
algebraic geometry settings to get this example; if one considers N
and X with Zarisky topology, it looks like the pullback in category
Top (of topological spaces) is also not open.
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Fortunately, this impression is wrong. (Fortunately, since it is easy
to prove that in Top any open mapping is universally open, and I
suspected that even my ability to handle Top was under suspicion. ;-)
The difference between these two categories is the topology on N'. If
one does a pullback to N in category Top, one gets a topological space
N'' with the same points as on N', but with weakier topology (as usual
when one compares inverse limits in categories of schemes and of
topological spaces). It is "pure Zariski" (a proper subset is closed
iff it is finite), thus there is only 1 irreducible component - it is
not a disjoint union.
Yours,
Ilya |
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