## 16.12 The main theorem

In this section we wrap up the discussion.

Proof. By Lemma 16.8.4 it suffices to prove this for $k \to \Lambda$ where $\Lambda$ is Noetherian and geometrically regular over $k$. Let $k \to A \to \Lambda$ be a factorization with $A$ a finite type $k$-algebra. It suffices to construct a factorization $A \to B \to \Lambda$ with $B$ of finite type such that $\mathfrak h_ B = \Lambda$, see Lemma 16.2.8. Hence we may perform Noetherian induction on the ideal $\mathfrak h_ A$. Pick a prime $\mathfrak q \supset \mathfrak h_ A$ such that $\mathfrak q$ is minimal over $\mathfrak h_ A$. It now suffices to resolve $k \to A \to \Lambda \supset \mathfrak q$ (as defined in the text following Situation 16.9.1). If the characteristic of $k$ is zero, this follows from Lemma 16.10.3. If the characteristic of $k$ is $p > 0$, this follows from Lemma 16.11.4. $\square$

Comment #6666 by 蒙古上单 on

I have a question: (Maybe stupid): Is any formally etale ring maps of Notherian rings a filtered colimt of etale ring maps?

Comment #6670 by on

Let $A = \mathbf{F}_p[x]_{(x)}$. The map $A \to A^\wedge$ is formally etale, see Lemma 109.43.4. But since the fraction field extension is nonalgebraic, the ring $A^\wedge$ cannot be a filtered colimit of etale $A$-algebras.

Comment #6674 by 蒙古上单 on

Thank you very much!

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