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The Stacks project

Lemma 15.35.2. 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/F/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 15.35.1. 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 15.34.2 and the lower horizontal by Proposition 15.35.1 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 10.128.2 and 10.106.3. \square


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