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I am reading Hartshorne's book of Algebraic Geometry. I am stuck in understanding why quotient field of the local domain $O_p$ (where $O_p$ denotes the ring of regular functions at a point $p$ in affine variety $Y$) is the field of rational functions $K(Y)$.

I need some help to understand this. Thanks

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  • $\begingroup$ He proves this, doesn't he? What part is unclear? $\endgroup$ – Hoot Jul 14 '16 at 17:16
  • $\begingroup$ On page 17 he wrote every rational function in K(Y) is in some O_p. How it gives K(Y) as quotient field of O_p is my question $\endgroup$ – user185640 Jul 14 '16 at 17:33
  • $\begingroup$ Everything is taking place inside of the field $K(Y)$. At this point he has established that all the $A(Y) \subseteq \mathcal O_P$ have the same quotient field, let's call it $L$, inside $K(Y)$. The only remaining worry is that $K(Y)$ might be strictly bigger than $L$. $\endgroup$ – Hoot Jul 14 '16 at 17:53
  • $\begingroup$ Exactly i agree with you $\endgroup$ – user185640 Jul 14 '16 at 17:54
  • $\begingroup$ Do you agree that he resolves this issue? He takes any element of $K(Y)$ and shows that it lies in some $\mathcal O_P \subseteq L$. $\endgroup$ – Hoot Jul 14 '16 at 17:59
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For a domain $A$, the quotient field of $A$ is isomorphic to the quotient field of $A_{\mathfrak{p}}$ for any prime $\mathfrak{p}$. This is because taking the quotient field of a domain is simply localizing at the prime ideal $(0)$, and in the case above, localizing at $\mathfrak{p}$ then localizing at $(0)_{\mathfrak{p}}$ is the same as just localizing at $(0)$.

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  • $\begingroup$ We know O_p is subring of the field K(Y), but how the localization of O_p gives K(Y) to be O _p 's quotient field is not clear $\endgroup$ – user185640 Jul 14 '16 at 17:12
  • $\begingroup$ Without even considering properties of localization, I think you just stated another way to see the answer. $A_{\mathfrak{p}}$ is a subring of $k$, the quotient field pf $A$. Thus its quotient field is contained in $k$. Try to prove the other containment. $\endgroup$ – basket Jul 14 '16 at 17:16
  • $\begingroup$ Asking this question basically i asked about the reverse containment that you asked me to try $\endgroup$ – user185640 Jul 14 '16 at 17:35

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