# The Stacks Project

## Tag 07Z3

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 16332–16341 (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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