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Let $R = \bigoplus_{d \in \mathbb{N}_0}R^{(d)} $ be a graded ring and for homogeneous ideals $I$, let $V_{proj\;R}(I) = \{p \supseteq I \mid p \text{ is a homogeneous prime ideal and } p \nsupseteq R_+\}$ where $R_+ = \bigoplus_{d \in \mathbb{N}} R^{(d)}$ is the irrelevant ideal. I want to show that $V_{proj\;R}(I) = V_{proj\;R}(I \cap R_+)$.

I have already shown that $I = \bigoplus_{d \in \mathbb{N}_0} I \cap R^{(d)}$ holds, but I currently fail to see why the grade zero elements are uniquely determined by the elements of higher grade and am therefore looking for hints on this question.

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    $\begingroup$ If P is a prime ideal and $I\cap J\subseteq P$ then what can conclude? Furthermore, what if $J$ is not contained in P? $\endgroup$ – user26857 Dec 7 '18 at 16:32
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As user26857 hinted at, if $I \cap R_+ \subset \mathfrak{p}$, then also $I \cdot R_+ \subset \mathfrak{p}$. Hence either $I \subset \mathfrak{p}$ or $R_+ \subset \mathfrak{p}$, because $\mathfrak{p}$ is prime. The latter is excluded for all primes in $V_{\text{proj }R}(I \cap R_+)$, so we see that $\mathfrak{p} \in V(I) \Leftrightarrow \mathfrak{p} \in V(I \cap R_+)$.

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  • $\begingroup$ Thanks, I found this out too after reading his comment. I really missed this one detail. $\endgroup$ – PLO Dec 7 '18 at 20:16

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