The Stacks project

Lemma 15.34.2. Let $k$ be a field of characteristic $p > 0$. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. Assume $A$ is geometrically regular over $k$. Let $k \subset F \subset K$ be a finitely generated subextension. Let $\varphi : k[y_1, \ldots , y_ m] \to A$ be a $k$-algebra map such that $y_ i$ maps to an element of $F$ in $K$ and such that $\text{d}y_1, \ldots , \text{d}y_ m$ map to a basis of $\Omega _{F/k}$. Set $\mathfrak p = \varphi ^{-1}(\mathfrak m)$. Then

\[ k[y_1, \ldots , y_ m]_\mathfrak p \to A \]

is flat and $A/\mathfrak pA$ is regular.

Proof. Set $A_0 = k[y_1, \ldots , y_ m]_\mathfrak p$ with maximal ideal $\mathfrak m_0$ and residue field $K_0$. Note that $\Omega _{A_0/k}$ is free of rank $m$ and $\Omega _{A_0/k} \otimes K_0 \to \Omega _{K_0/k}$ is an isomorphism. It is clear that $A_0$ is geometrically regular over $k$. Hence $H_1(L_{K_0/k}) \to \mathfrak m_0/\mathfrak m_0^2$ is an isomorphism, see Proposition 15.34.1. Now consider

\[ \xymatrix{ H_1(L_{K_0/k}) \otimes K \ar[d] \ar[r] & \mathfrak m_0/\mathfrak m_0^2 \otimes K \ar[d] \\ H_1(L_{K/k}) \ar[r] & \mathfrak m/\mathfrak m^2 } \]

Since the left vertical arrow is injective by Lemma 15.33.2 and the lower horizontal by Proposition 15.34.1 we conclude that the right vertical one is too. Hence a regular system of parameters in $A_0$ maps to part of a regular system of parameters in $A$. We win by Algebra, Lemmas 10.127.2 and 10.105.3. $\square$


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