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Tag 07Z3

Chapter 10: Commutative Algebra > Section 10.69: Blow up algebras

Lemma 10.69.2. Let $R$ be a ring, $I \subset R$ an ideal, and $a \in I$. Let $R' = R[\frac{I}{a}]$ be the affine blowup algebra. Then

  1. the image of $a$ in $R'$ is a nonzerodivisor,
  2. $IR' = aR'$, and
  3. $(R')_a = R_a$.

Proof. Immediate from the description of $R[\frac{I}{a}]$ above. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 16362–16371 (see updates for more information).

    \begin{lemma}
    \label{lemma-affine-blowup}
    Let $R$ be a ring, $I \subset R$ an ideal, and $a \in I$.
    Let $R' = R[\frac{I}{a}]$ be the affine blowup algebra. Then
    \begin{enumerate}
    \item the image of $a$ in $R'$ is a nonzerodivisor,
    \item $IR' = aR'$, and
    \item $(R')_a = R_a$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Immediate from the description of $R[\frac{I}{a}]$ above.
    \end{proof}

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