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Are inclusions of open subspaces monomorphisms of ringed spaces?
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seldnplan@gmail.com
science forum beginner


Joined: 30 Dec 2005
Posts: 8

PostPosted: Fri Jun 23, 2006 2:30 pm    Post subject: Are inclusions of open subspaces monomorphisms of ringed spaces? Reply with quote

It is true in the category of schemes that open immersions are
monomorphisms, but I can only prove this fact from the fact that for
any ring R and any element f in R the localization map R\to R_f is an
epimorphism. Can this be proved in general for ringed spaces? More
precisely, if I have a ringed space (X,O_X) and an open subspace
(U,O_U), where O_U is just the restriction of O_X to U, will the
natural inclusion U\subset X be a monomorphism of ringed spaces?

Thanks,
Keerthi
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Lukas-Fabian Moser
science forum beginner


Joined: 08 Apr 2005
Posts: 4

PostPosted: Fri Jun 23, 2006 7:30 pm    Post subject: Re: Are inclusions of open subspaces monomorphisms of ringed spaces? Reply with quote

Hallo,

On Fri, 23 Jun 2006 14:30:07 +0000 (UTC), seldnplan@gmail.com wrote:

Quote:
It is true in the category of schemes that open immersions are
monomorphisms, but I can only prove this fact from the fact that for
any ring R and any element f in R the localization map R\to R_f is an
epimorphism. Can this be proved in general for ringed spaces? More
precisely, if I have a ringed space (X,O_X) and an open subspace
(U,O_U), where O_U is just the restriction of O_X to U, will the
natural inclusion U\subset X be a monomorphism of ringed spaces?

What do you mean by "monomorphism"? If it's just that open immersions
can be cancelled from the left (i o f = i o g => f = g for an open
immersion i), this is of course true. One can see this by writing down
the definition of a composition of morphisms of ringed spaces and of
the (to a topological open immersion) associated morphism of sheaves;
but when doing so, one can just as well go on and prove the following
universal mapping property of open immersions: if U -> X is an open
immersion then every morphism f: W -> X with f^(-1)(U) = W factors
uniquely through U.

Lukas
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