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seldnplan@gmail.com
science forum beginner
Joined: 30 Dec 2005
Posts: 8
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 Posted: Mon Jun 12, 2006 12:46 pm Post subject:
Quasicoherent sheaves
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This is just idle musing, but is there an (easy?) example of a ringed
space whose quasicoherent sheaves are not stable under extensions? |
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Jannick Asmus
science forum Guru
Joined: 25 Mar 2005
Posts: 312
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| Posted: Tue Jun 13, 2006 8:38 pm Post subject:
Re: Quasicoherent sheaves
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On 12.06.2006 14:46, seldnplan@gmail.com wrote:
| Quote: | This is just idle musing, but is there an (easy?) example of a ringed
space whose quasicoherent sheaves are not stable under extensions?
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I must admit that I only know the notion of a quasi-coherent sheaf on
schemes which are very special ringed spaces. This is described, e.g.,
in Hartshorne's book on algebraic geometry.
If you could give a definition of a sheaf to be quasi-coherent, someone
or me could try to find an answer to your question.
Best wishes,
J. |
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Ryan Reich
science forum Guru Wannabe
Joined: 21 May 2005
Posts: 120
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| Posted: Wed Jun 14, 2006 5:59 pm Post subject:
Re: Quasicoherent sheaves
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On Tue, 13 Jun 2006 22:38:08 +0200, Jannick Asmus <jannick.news@web.de> wrote:
| Quote: | On 12.06.2006 14:46, seldnplan@gmail.com wrote:
This is just idle musing, but is there an (easy?) example of a ringed
space whose quasicoherent sheaves are not stable under extensions?
I must admit that I only know the notion of a quasi-coherent sheaf on
schemes which are very special ringed spaces. This is described, e.g.,
in Hartshorne's book on algebraic geometry.
If you could give a definition of a sheaf to be quasi-coherent, someone
or me could try to find an answer to your question.
Best wishes,
J.
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In EGA I, Grothendieck defines (at the beginning) a more general notion of
quasicoherent sheaves, which are those shaves that are locally the cokernel of
a map of free sheaves (that is, which have "local presentations"). Then he
proves, much later in the book, that this coincides with the other definition
for schemes.
--
Ryan Reich
ryan.reich@gmail.com
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