## Tag `00QO`

Chapter 10: Commutative Algebra > Section 10.126: Colimits and maps of finite presentation

Lemma 10.126.3. Let $\varphi : R \to S$ be a ring map. The following are equivalent

- $\varphi$ is of finite presentation,
- for every directed system $A_\lambda$ of $R$-algebras the map $$ \mathop{\rm colim}\nolimits_\lambda \mathop{\rm Hom}\nolimits_R(S, A_\lambda) \longrightarrow \mathop{\rm Hom}\nolimits_R(S, \mathop{\rm colim}\nolimits_\lambda A_\lambda) $$ is bijective, and
- for every directed system $A_\lambda$ of $R$-algebras the map $$ \mathop{\rm colim}\nolimits_\lambda \mathop{\rm Hom}\nolimits_R(S, A_\lambda) \longrightarrow \mathop{\rm Hom}\nolimits_R(S, \mathop{\rm colim}\nolimits_\lambda A_\lambda) $$ is surjective.

Proof.Assume (1) and write $S = R[x_1, \ldots, x_n] / (f_1, \ldots, f_m)$. Let $A = \mathop{\rm colim}\nolimits A_\lambda$. Observe that an $R$-algebra homomorphism $S \to A$ or $S \to A_\lambda$ is determined by the images of $x_1, \ldots, x_n$. Hence it is clear that $\mathop{\rm colim}\nolimits_\lambda \mathop{\rm Hom}\nolimits_R(S, A_\lambda) \to \mathop{\rm Hom}\nolimits_R(S, A)$ is injective. To see that it is surjective, let $\chi : S \to A$ be an $R$-algebra homomorphism. Then each $x_i$ maps to some element in the image of some $A_{\lambda_i}$. We may pick $\mu \geq \lambda_i$, $i = 1, \ldots, n$ and assume $\chi(x_i)$ is the image of $y_i \in A_\mu$ for $i = 1, \ldots, n$. Consider $z_j = f_j(y_1, \ldots, y_n) \in A_\mu$. Since $\chi$ is a homomorphism the image of $z_j$ in $A = \mathop{\rm colim}\nolimits_\lambda A_\lambda$ is zero. Hence there exists a $\mu_j \geq \mu$ such that $z_j$ maps to zero in $A_{\mu_j}$. Pick $\nu \geq \mu_j$, $j = 1, \ldots, m$. Then the images of $z_1, \ldots, z_m$ are zero in $A_\nu$. This exactly means that the $y_i$ map to elements $y'_i \in A_\nu$ which satisfy the relations $f_j(y'_1, \ldots, y'_n) = 0$. Thus we obtain a ring map $S \to A_\nu$. This shows that (1) implies (2).It is clear that (2) implies (3). Assume (3). By Lemma 10.126.2 we may write $S = \mathop{\rm colim}\nolimits_\lambda S_\lambda$ with $S_\lambda$ of finite presentation over $R$. Then the identity map factors as $$ S \to S_\lambda \to S $$ for some $\lambda$. This implies that $S$ is finitely presented over $S_\lambda$ by Lemma 10.6.2 part (4) applied to $S \to S_\lambda \to S$. Applying part (2) of the same lemma to $R \to S_\lambda \to S$ we conclude that $S$ is of finite presentation over $R$. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 30504–30524 (see updates for more information).

```
\begin{lemma}
\label{lemma-characterize-finite-presentation}
Let $\varphi : R \to S$ be a ring map. The following are equivalent
\begin{enumerate}
\item $\varphi$ is of finite presentation,
\item for every directed system $A_\lambda$ of $R$-algebras
the map
$$
\colim_\lambda \Hom_R(S, A_\lambda) \longrightarrow
\Hom_R(S, \colim_\lambda A_\lambda)
$$
is bijective, and
\item for every directed system $A_\lambda$ of $R$-algebras
the map
$$
\colim_\lambda \Hom_R(S, A_\lambda) \longrightarrow
\Hom_R(S, \colim_\lambda A_\lambda)
$$
is surjective.
\end{enumerate}
\end{lemma}
\begin{proof}
Assume (1) and write $S = R[x_1, \ldots, x_n] / (f_1, \ldots, f_m)$.
Let $A = \colim A_\lambda$. Observe that an $R$-algebra homomorphism
$S \to A$ or $S \to A_\lambda$ is determined by the images of
$x_1, \ldots, x_n$. Hence it is clear that
$\colim_\lambda \Hom_R(S, A_\lambda) \to \Hom_R(S, A)$
is injective. To see that it is surjective, let $\chi : S \to A$
be an $R$-algebra homomorphism. Then each
$x_i$ maps to some element in the image of some $A_{\lambda_i}$.
We may pick $\mu \geq \lambda_i$, $i = 1, \ldots, n$ and
assume $\chi(x_i)$ is the image of $y_i \in A_\mu$ for
$i = 1, \ldots, n$. Consider $z_j = f_j(y_1, \ldots, y_n) \in A_\mu$.
Since $\chi$ is a homomorphism the image of $z_j$ in
$A = \colim_\lambda A_\lambda$ is zero. Hence there exists a
$\mu_j \geq \mu$ such that $z_j$ maps to zero in $A_{\mu_j}$.
Pick $\nu \geq \mu_j$, $j = 1, \ldots, m$. Then the
images of $z_1, \ldots, z_m$ are zero in $A_\nu$. This
exactly means that the $y_i$ map to elements
$y'_i \in A_\nu$ which satisfy the relations $f_j(y'_1, \ldots, y'_n) = 0$.
Thus we obtain a ring map $S \to A_\nu$. This shows that
(1) implies (2).
\medskip\noindent
It is clear that (2) implies (3). Assume (3).
By Lemma \ref{lemma-ring-colimit-fp} we may write
$S = \colim_\lambda S_\lambda$ with $S_\lambda$
of finite presentation over $R$. Then the identity map
factors as
$$
S \to S_\lambda \to S
$$
for some $\lambda$. This implies that $S$
is finitely presented over $S_\lambda$ by
Lemma \ref{lemma-compose-finite-type} part (4)
applied to $S \to S_\lambda \to S$. Applying part (2) of the same
lemma to $R \to S_\lambda \to S$ we conclude that $S$ is of finite
presentation over $R$.
\end{proof}
```

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