Prove that dim Spec A = dim Spec A/p + dim Spec A_p

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dataislyfe
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Prove that dim Spec A = dim Spec A/p + dim Spec A_p

Postby dataislyfe » Thu Feb 22, 2018 10:38 am UTC

I'm trying to work on this exercise:

Let k be an infinite field and suppose A is a k-finitely generated integral domain (hence,
Spec A is irreducible. Let p be a prime ideal of A and and let A_p be the localization of
A at p. Prove that dim Spec A = dim Spec A/p + dim Spec A_p.

I was wondering if you can prove this using transcendence degree + Noether normalization? If not,
I'm not sure how to proceed so some hints would be very helpful!

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doogly
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Re: Prove that dim Spec A = dim Spec A/p + dim Spec A_p

Postby doogly » Fri Feb 23, 2018 1:37 am UTC

This stinks like something with a two line proof using categories.
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Demki
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Re: Prove that dim Spec A = dim Spec A/p + dim Spec A_p

Postby Demki » Fri Feb 23, 2018 11:24 am UTC

doogly wrote:This stinks like something with a two line proof using categories.

Obligatory reference


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