This is related to Ueno, Algebraic Geometry 2, Example 5.28 of Chpt 5.
Let $R=k[x_1,\dots, x_n]$. Let $I=(x_1,\dots, x_n)$. Form blow up algebra $S=R\oplus I\oplus I^2\oplus\dots$ which is graded $R$ algebra. There is canonical $R\to S$ embedding into $0-$th degree component. This induces $\pi: Proj(S)\to Spec(R)$ map. Consider $U=Spec(R)-\{(0,\dots, 0)\}$. I will denote $S$ as the associated sheaf of $S$ as $O_X$ algebra where $X=Spec(R)$.
The book says $S|_U\cong\oplus_d O_U^{\otimes d}=O_U[T]$.
$\textbf{Q:}$ Why $S|_U\cong\oplus_d O_U^{\otimes d}=O_U[T]$ is obvious? I could not see this totally obvious unless I pick $U=D(x_i)\subset Spec(R)$ which covers $U$. Hence I see isomorphism easily as all $I|_{D(x_i)}=R_{x_i}$. It seems there is some mechanism that the book sees this obvious as there is no indication for the implication.