The Stacks project

Lemma 10.127.2. Let $R \to A$ be a ring map. There exists a directed system $A_\lambda $ of $R$-algebras of finite presentation such that $A = \mathop{\mathrm{colim}}\nolimits _\lambda A_\lambda $. If $A$ is of finite type over $R$ we may arrange it so that all the transition maps in the system of $A_\lambda $ are surjective.

Proof. The first proof is that this follows from Lemma 10.127.1 and Categories, Lemma 4.21.5.

Second proof. Compare with the proof of Lemma 10.11.3. Consider any finite subset $S \subset A$, and any finite collection of polynomial relations $E$ among the elements of $S$. So each $s \in S$ corresponds to $x_ s \in A$ and each $e \in E$ consists of a polynomial $f_ e \in R[X_ s; s\in S]$ such that $f_ e(x_ s) = 0$. Let $A_{S, E} = R[X_ s; s\in S]/(f_ e; e\in E)$ which is a finitely presented $R$-algebra. There are canonical maps $A_{S, E} \to A$. If $S \subset S'$ and if the elements of $E$ correspond, via the map $R[X_ s; s \in S] \to R[X_ s; s\in S']$, to a subset of $E'$, then there is an obvious map $A_{S, E} \to A_{S', E'}$ commuting with the maps to $A$. Thus, setting $\Lambda $ equal the set of pairs $(S, E)$ with ordering by inclusion as above, we get a directed partially ordered set. It is clear that the colimit of this directed system is $A$.

For the last statement, suppose $A = R[x_1, \ldots , x_ n]/I$. In this case, consider the subset $\Lambda ' \subset \Lambda $ consisting of those systems $(S, E)$ above with $S = \{ x_1, \ldots , x_ n\} $. It is easy to see that still $A = \mathop{\mathrm{colim}}\nolimits _{\lambda ' \in \Lambda '} A_{\lambda '}$. Moreover, the transition maps are clearly surjective. $\square$

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