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I am trying to learn about scheme theoretic algebraic geometry, because I actually want to study the basics of interseciton theory. I stumbled across the term "local ring $\mathcal{O}_{V,X}$ of a scheme $X$ along a closed subscheme/subvariety/primedivisor $V$" many times but I just can't find a definition of this term in my textbooks on schemes. (I have looked at Bosch, Eisenbud & Harris, FOAG by Vakil,...)

I am sure they explain it but I just don't see it. Could anybody help me out here?

Teddyboer
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    $V$ as a subvariety has a unique generic point. Consider the local ring with respect to that point. – awllower Feb 13 '20 at 11:10
  • So every irreducible closed subset $V$ of a scheme admits a unique generic point $x$. The only local ring I can thing of would be the stalk $\mathcal{O}{X,x}$ at $x$. Is this what is meant by $\mathcal{O}{V,X}$? Does this also have a different name? – Teddyboer Feb 13 '20 at 12:33
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    Yes, see this question. Yes, I think it is what is meant by $\mathcal O_{V, X}$. I don't know of other names though. BTW, a space in which every irreducible closed subspace has a unique generic point is said to be sober. Consider the lemma from Stacks Project 01IS. – awllower Feb 13 '20 at 15:41
  • @awllower You are correct - would you care to record your comments as an answer? – KReiser Feb 13 '20 at 17:18

1 Answers1

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By this question, or by Stacks Project Lemma 01IS, every irreducible closed subspace of a scheme has a unique generic point, i.e. the topological space underlying a scheme is sober. I think by the local ring $\mathcal O_{V, X}$ of a scheme $X$ along a sub-variety $V$ is meant the local ring $\mathcal O_{X, x}$ of $X$ with respect to the unique generic point $x$ of $V$.

Hope this helps.

awllower
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