Let $f \in k[x_0,\ldots,x_n]$ be a homogeneous irreducible polynomial.And let $X=\mathbb{P}^n - \mathcal{Z}(f)$.It is known that $X$ is isomorphic to an affine variety. Is it true that its coordinate ring $A(X)$ is isomorphic to $k[x_0,\ldots,x_n]_{(f)}$ ?
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If that is your notation for the degree-zero part of the localization of $k[x_0,\ldots,x_n]$ in $f$, then this is true. A way to see it is by applying the $d$-uple (veronese) embedding. That way, you can assume $f$ to be linear, i.e. you may assume that it is one of the $x_i$. In this case, your statement is just the well-known fact that the standard open sets $U_i=\mathbb{P}^n\setminus\mathcal{Z}(x_i)$ are isomorphic to $\mathbb{A}^n$. |
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