# Quotient field of ring of regular functions at some point in affine variety is the field of rational functions on the variety

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

• He proves this, doesn't he? What part is unclear? – Hoot Jul 14 '16 at 17:16
• 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 – user185640 Jul 14 '16 at 17:33
• 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 $L$. – Hoot Jul 14 '16 at 17:53
• Exactly i agree with you – user185640 Jul 14 '16 at 17:54
• Do you agree that he resolves this issue? He takes any element of $K(Y)$ and shows that it lies in some $\mathcal O_P \subseteq L$. – Hoot Jul 14 '16 at 17:59

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 $(0)$.
• 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. – basket Jul 14 '16 at 17:16