Theorem 16.12.1 (Popescu). Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps.
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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.