Theorem 16.12.1 (Popescu). Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps.

## 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.
$\square$

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