Definition 111.27.3. Let $R = \oplus _{d \geq 0} R_ d$ be a graded ring, let $f\in R_ d$ and assume that $d \geq 1$. We define *$R_{(f)}$* to be the subring of $R_ f$ consisting of elements of the form $r/f^ n$ with $r$ homogeneous and $\deg (r) = nd$. Furthermore, we define

Finally, for a homogeneous ideal $I \subset R$ we define $V_{+}(I) = V(I) \cap \text{Proj}(R)$.

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