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Tag 0BBI

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

Lemma 10.69.5. Let $R$ be a ring, $I \subset R$ an ideal, $a \in I$, and $f \in R$. Set $R' = R[\frac{I}{a}]$ and $R'' = R[\frac{fI}{fa}]$. Then there is a surjective $R$-algebra map $R' \to R''$ whose kernel is the set of $f$-power torsion elements of $R'$.

Proof. The map is given by sending $x/a^n$ for $x \in I^n$ to $f^nx/(fa)^n$. It is straightforward to check this map is well defined and surjective. Since $af$ is a nonzero divisor in $R''$ (Lemma 10.69.2) we see that the set of $f$-power torsion elements are mapped to zero. Conversely, if $x \in R'$ and $f^n x \not = 0$ for all $n > 0$, then $(af)^n x \not = 0$ for all $n$ as $a$ is a nonzero divisor in $R'$. It follows that the image of $x$ in $R''$ is not zero by the description of $R''$ following Definition 10.69.1. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 16388–16394 (see updates for more information).

    \begin{lemma}
    \label{lemma-blowup-add-principal}
    Let $R$ be a ring, $I \subset R$ an ideal, $a \in I$, and $f \in R$.
    Set $R' = R[\frac{I}{a}]$ and $R'' = R[\frac{fI}{fa}]$. Then
    there is a surjective $R$-algebra map $R' \to R''$ whose kernel
    is the set of $f$-power torsion elements of $R'$.
    \end{lemma}
    
    \begin{proof}
    The map is given by sending $x/a^n$ for $x \in I^n$ to $f^nx/(fa)^n$.
    It is straightforward to check this map is well defined and surjective.
    Since $af$ is a nonzero divisor in $R''$
    (Lemma \ref{lemma-affine-blowup}) we see that the set of $f$-power
    torsion elements are mapped to zero. Conversely, if $x \in R'$
    and $f^n x \not = 0$ for all $n > 0$, then $(af)^n x \not = 0$
    for all $n$ as $a$ is a nonzero divisor in $R'$. It follows
    that the image of $x$ in $R''$ is not zero by the description of
    $R''$ following Definition \ref{definition-blow-up}.
    \end{proof}

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