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Tag 00JU

Chapter 10: Commutative Algebra > Section 10.56: Proj of a graded ring

Lemma 10.56.8. Let $S$ be a graded ring.

  1. Any minimal prime of $S$ is a homogeneous ideal of $S$.
  2. Given a homogeneous ideal $I \subset S$ any minimal prime over $I$ is homogeneous.

Proof. The first assertion holds because the prime $\mathfrak q$ constructed in Lemma 10.56.7 satisfies $\mathfrak q \subset \mathfrak p$. The second because we may consider $S/I$ and apply the first part. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 13048–13056 (see updates for more information).

    \begin{lemma}
    \label{lemma-graded-ring-minimal-prime}
    Let $S$ be a graded ring.
    \begin{enumerate}
    \item Any minimal prime of $S$ is a homogeneous ideal of $S$.
    \item Given a homogeneous ideal $I \subset S$ any minimal
    prime over $I$ is homogeneous.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    The first assertion holds because the prime $\mathfrak q$ constructed in
    Lemma \ref{lemma-smear-out} satisfies $\mathfrak q \subset \mathfrak p$.
    The second because we may consider $S/I$ and apply the first part.
    \end{proof}

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