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 Post subject: the ideal of a varietyPosted: Tue, 15 May 2012 16:25:09 UTC
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Joined: Mon, 14 May 2012 20:59:26 UTC
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Can someone explain me how can i find I(V(I)) if I is an ideal?
I know the definition of the ideal of a variety:
For a variety V is

but when I find the points of th set V(I), then, i don't know how find I(V(I))

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 Post subject: Re: the ideal of a varietyPosted: Tue, 15 May 2012 16:48:04 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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guachu wrote:
Can someone explain me how can i find I(V(I)) if I is an ideal?
I know the definition of the ideal of a variety:
For a variety V is

but when I find the points of th set V(I), then, i don't know how find I(V(I))

Recall Hilbert's Nullstellensatz.

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 Post subject: Re: the ideal of a varietyPosted: Tue, 15 May 2012 18:11:29 UTC
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Joined: Mon, 14 May 2012 20:59:26 UTC
Posts: 35
But, for example, you have the ideal and you want to find V(I) and I(V(I)).
V(I) is easy i think, because you have to look for the points so that that are the points . But what would you do to calculate I(V(I))?

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 Post subject: Re: the ideal of a varietyPosted: Tue, 15 May 2012 21:02:09 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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guachu wrote:
But, for example, you have the ideal and you want to find V(I) and I(V(I)).
V(I) is easy i think, because you have to look for the points so that that are the points . But what would you do to calculate I(V(I))?

Well certainly , indeed the Nulstellensatz tells you that . Now you just need to compute which is...

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 Post subject: Re: the ideal of a varietyPosted: Tue, 15 May 2012 22:39:46 UTC
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Joined: Mon, 14 May 2012 20:59:26 UTC
Posts: 35
I thought it necessary to work in an algebraically closed field to apply Nulstellensatz, and if the ideal is in R[x,y], R isn't algebraically closed field...
And... why you know that ?? I know that is true, but the reverse... why?

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 Post subject: Re: the ideal of a varietyPosted: Wed, 16 May 2012 01:07:29 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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guachu wrote:
I thought it necessary to work in an algebraically closed field to apply Nulstellensatz, and if the ideal is in R[x,y], R isn't algebraically closed field...
And... why you know that ?? I know that is true, but the reverse... why?

Oops, it seems I accidently wrote those backwards. Aplogies, you are correct.

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 Post subject: Re: the ideal of a varietyPosted: Wed, 16 May 2012 05:41:40 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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guachu wrote:
I thought it necessary to work in an algebraically closed field to apply Nulstellensatz, and if the ideal is in R[x,y], R isn't algebraically closed field...
And... why you know that ?? I know that is true, but the reverse... why?

Then look at the version of Nullstellensatz over Jacobson rings. Of course, you should know about schemes before you try out the algebraic-geometry equivalent.

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