## Tag `00R2`

Chapter 10: Commutative Algebra > Section 10.6: Ring maps of finite type and of finite presentation

Lemma 10.6.3. Let $R \to S$ be a ring map of finite presentation. For any surjection $\alpha : R[x_1, \ldots, x_n] \to S$ the kernel of $\alpha$ is a finitely generated ideal in $R[x_1, \ldots, x_n]$.

Proof.Write $S = R[y_1, \ldots, y_m]/(f_1, \ldots, f_k)$. Choose $g_i \in R[y_1, \ldots, y_m]$ which are lifts of $\alpha(x_i)$. Then we see that $S = R[x_i, y_j]/(f_l, x_i - g_i)$. Choose $h_j \in R[x_1, \ldots, x_n]$ such that $\alpha(h_j)$ corresponds to $y_j \bmod (f_1, \ldots, f_k)$. Consider the map $\psi : R[x_i, y_j] \to R[x_i]$, $x_i \mapsto x_i$, $y_j \mapsto h_j$. Then the kernel of $\alpha$ is the image of $(f_l, x_i - g_i)$ under $\psi$ and we win. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 557–562 (see updates for more information).

```
\begin{lemma}
\label{lemma-finite-presentation-independent}
Let $R \to S$ be a ring map of finite presentation.
For any surjection $\alpha : R[x_1, \ldots, x_n] \to S$ the
kernel of $\alpha$ is a finitely generated ideal in $R[x_1, \ldots, x_n]$.
\end{lemma}
\begin{proof}
Write $S = R[y_1, \ldots, y_m]/(f_1, \ldots, f_k)$.
Choose $g_i \in R[y_1, \ldots, y_m]$ which are lifts
of $\alpha(x_i)$. Then we see that $S = R[x_i, y_j]/(f_l, x_i - g_i)$.
Choose $h_j \in R[x_1, \ldots, x_n]$ such that $\alpha(h_j)$
corresponds to $y_j \bmod (f_1, \ldots, f_k)$. Consider
the map $\psi : R[x_i, y_j] \to R[x_i]$, $x_i \mapsto x_i$,
$y_j \mapsto h_j$. Then the kernel of $\alpha$
is the image of $(f_l, x_i - g_i)$ under $\psi$ and we win.
\end{proof}
```

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