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Math Forum / Mathematics / General Topics / October 2008reduced scheme
Hi take an open and closed subset A of the topological space of a scheme X, then one can furnish it with the structure of an open subscheme of X by restricting the sheaf on X to A. On the other hand you can put the reduced structure on it to make it a closed subscheme of X. Are there two structures the same? Is this true if X is reduced (and some other good properties...) at least? Yours Alex On 30 Okt., 18:44, mynameisrab...@hotmail.com wrote: > Hi > [quoted text clipped - 7 lines] > Yours > Alex Can anybody give me a clue if this is true? On Oct 30, 3:55 pm, mynameisrab...@hotmail.com wrote: > On 30 Okt., 18:44, mynameisrab...@hotmail.com wrote: > [quoted text clipped - 11 lines] > > Can anybody give me a clue if this is true? Be careful about the distinction between a sheaf of abelian groups on a scheme, and a module over the scheme's structure sheaf; restricting a sheaf to a subscheme makes sense for a sheaf of abelian groups, but to restrict an O_X-module to A, you need to specify how the restricted sheaf is an O_A-module. If you write f for the inclusion of A into X, then f^{-1} restricts a sheaf on X to A but it's only defined on sheaves of abelian groups on X, while f^*, defined on O_X-modules, is what gives you an O_A-module. The two operations are different. I suspect a confusion about these two operations on two different kinds of sheaves is what is at the root of your question. On 30 Okt., 21:07, anonymous.rubbert...@yahoo.com wrote: > On Oct 30, 3:55 pm, mynameisrab...@hotmail.com wrote: > [quoted text clipped - 24 lines] > suspect a confusion about these two operations on two different kinds > of sheaves is what is at the root of your question. Thank you for your answer. I am only interested in the structure sheaf of X. So my question is: Let A be a open and closed subset of a scheme X. Does it make a difference to consider A as a closed subscheme of X with the reduced structure sheaf or as an open subscheme with the restriction of the structure sheaf of X? On Oct 30, 6:38 pm, mynameisrab...@hotmail.com wrote: > On 30 Okt., 21:07, anonymous.rubbert...@yahoo.com wrote: > [quoted text clipped - 32 lines] > with the reduced structure sheaf or as an open subscheme with the > restriction of the structure sheaf of X? Call A1 the scheme you get by restricting X to A as an open subset, and call A2 the scheme you get by inducing on A the structure of a reduced closed subscheme (and assume that X is reduced, for otherwise A1 and A2 are aleady different when X=A...) Can you describe the values at an open subset U of A of the structure sheafs of A1 and A2? -- m On 30 Okt., 23:13, Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com> wrote: > On Oct 30, 6:38 pm, mynameisrab...@hotmail.com wrote: > [quoted text clipped - 45 lines] > > -- m Ok, thanks, I think it works. Your A here is supposed to be an open and closed subset of X, but such subschemes A are, in a certain sense, scarce. Any integral scheme X has a generic point, a point which is in every open subset of X. If A is nonempty and open and closed in X, then A must contain the generic point, since A is open; but the complement of A is also open, so either the complement of A is empty, or the complement of A also contains the generic point. It cannot be true that both A and its complement contain the generic point; so either A is all of X, or A is empty. On 31 Okt., 17:15, anonymous.rubbert...@yahoo.com wrote: > Your A here is supposed to be an open and closed subset of X, but such > subschemes A are, in a certain sense, scarce. Any integral scheme X [quoted text clipped - 5 lines] > complement contain the generic point; so either A is all of X, or A is > empty. If the scheme X is not connected, say with two components, then each one is open and closed, right? On Oct 31, 2:15 pm, anonymous.rubbert...@yahoo.com wrote: > Your A here is supposed to be an open and closed subset of X, but such > subschemes A are, in a certain sense, scarce. Any integral scheme X [quoted text clipped - 5 lines] > complement contain the generic point; so either A is all of X, or A is > empty. There are plenty of reduced schemes which are not connected... Say... Spec k[x]/(x-x^2), for example. -- m On Oct 31, 12:20 pm, Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com> wrote: > On Oct 31, 2:15 pm, anonymous.rubbert...@yahoo.com wrote: > [quoted text clipped - 12 lines] > > -- m The ring k[x]/(x-x^2) has zero divisors, though, so Spec k[x]/(x-x^2) is not an integral scheme. If a scheme X is integral, then any affine subscheme of X is the prime spectrum of an integral domain--so it is connected. On Oct 31, 2:34 pm, anonymous.rubbert...@yahoo.com wrote: > On Oct 31, 12:20 pm, Mariano Suárez-Alvarez > [quoted text clipped - 21 lines] > If a scheme X is integral, then any affine subscheme of X is the prime > spectrum of an integral domain--so it is connected. The restriction to integral schemes was introduced by you. The OP is obviously interested in non-connected schemes. His question was: "are the structures of open subscheme and of reduced closed subscheme the same on a connected component?" -- m On Oct 31, 12:58 pm, Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com> wrote: > On Oct 31, 2:34 pm, anonymous.rubbert...@yahoo.com wrote: > [quoted text clipped - 31 lines] > > -- m Oops--I suddenly don't think my remarks were helpful at all. Sorry for making things more unclear. |
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