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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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