# Localization of Coordinate Rings: $\mathbb C[V_f] = \mathbb C[V]_f$.

Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb C[V]\setminus\{0\}$, then we can define the localization $$\mathbb C[V]_f = \left\{\,g\big/f^\ell \in \mathbb C(V)\,\big|\, g\in\mathbb C[V], \ell\ge 0\,\right\},$$ where $\mathbb C(V)$ denotes the field of fractions of $\mathbb C[V]$.

I want to proof that $\mathbb C[V]_f$ is the coordinate ring of the principal open subset $$V_f = \left\{\,p\in V\,\big|\, f(p)\neq 0\,\right\}.$$

We can see $V_f$ as an affine variety by identifying it with $$\widetilde{V_f} = \mathbf V(\mathbf I(V)+\langle gy-1\rangle) \subseteq \mathbb C^n\times \mathbb C,$$ where $y$ is $(n+1)$th coordinate, $g\in\mathbb C[x_1,\dots,x_n]$ represents $f\in\mathbb C[V]$ and the projection $\mathbb C^n\times \mathbb C\to\mathbb C^n$ maps $\widetilde{V_f}$ bijectively onto $V_f$. Then \begin{align} \mathbb C[V_f] &\cong \mathbb C[\widetilde{V_f}] = \mathbb C[x_1,\dots,x_n,y]\big/(\mathbf I(V)+\langle gy-1\rangle)\\ &\cong \mathbb C\left[x_1,\dots,x_n,1\big/g\right]\big/\mathbf I(V)\\ &\cong \left(\mathbb C[V]\right)\left[1\big/f\right] \cong \mathbb C[V]_f. \end{align} Is this reasoning correct? I'm not sure everything done in the last chain of isomorphisms is rigorous.

## 1 Answer

Yes it's correct.
For details see, for example, these notes definition 1.13 and the example right before definition 1.16.

• Why $A(X)_{f} \cong K[x_{1}, \dots, x_{n},y]/J$ in the notes? I know that $R_{f} \cong R[X]/(xf-1)$ in general, but I don't see how this finishes. Commented Dec 16, 2019 at 23:13