Lemma 10.135.12. Let
\xymatrix{ B & S \ar[l] \\ A \ar[u] & R \ar[l] \ar[u] }
be a commutative square of local rings. Assume
R and \overline{S} = S/\mathfrak m_ R S are regular local rings,
A = R/I and B = S/J for some ideals I, J,
J \subset S and \overline{J} = J/\mathfrak m_ R \cap J \subset \overline{S} are generated by regular sequences, and
A \to B and R \to S are flat.
Then I is generated by a regular sequence.
Proof.
Set \overline{B} = B/\mathfrak m_ RB = B/\mathfrak m_ AB so that \overline{B} = \overline{S}/\overline{J}. Let f_1, \ldots , f_{\overline{c}} \in J be elements such that \overline{f}_1, \ldots , \overline{f}_{\overline{c}} \in \overline{J} form a regular sequence generating \overline{J}. Note that \overline{c} = \dim (\overline{S}) - \dim (\overline{B}), see Lemma 10.135.6. By Lemma 10.99.3 the ring S/(f_1, \ldots , f_{\overline{c}}) is flat over R. Hence S/(f_1, \ldots , f_{\overline{c}}) + IS is flat over A. The map S/(f_1, \ldots , f_{\overline{c}}) + IS \to B is therefore a surjection of finite S/IS-modules flat over A which is an isomorphism modulo \mathfrak m_ A, and hence an isomorphism by Lemma 10.99.1. In other words, J = (f_1, \ldots , f_{\overline{c}}) + IS.
By Lemma 10.135.6 again the ideal J is generated by a regular sequence of c = \dim (S) - \dim (B) elements. Hence J/\mathfrak m_ SJ is a vector space of dimension c. By the description of J above there exist g_1, \ldots , g_{c - \overline{c}} \in I such that J is generated by f_1, \ldots , f_{\overline{c}}, g_1, \ldots , g_{c - \overline{c}} (use Nakayama's Lemma 10.20.1). Consider the ring A' = R/(g_1, \ldots , g_{c - \overline{c}}) and the surjection A' \to A. We see from the above that B = S/(f_1, \ldots , f_{\overline{c}}, g_1, \ldots , g_{c - \overline{c}}) is flat over A' (as S/(f_1, \ldots , f_{\overline{c}}) is flat over R). Hence A' \to B is injective (as it is faithfully flat, see Lemma 10.39.17). Since this map factors through A we get A' = A. Note that \dim (B) = \dim (A) + \dim (\overline{B}), and \dim (S) = \dim (R) + \dim (\overline{S}), see Lemma 10.112.7. Hence c - \overline{c} = \dim (R) -\dim (A) by elementary algebra. Thus I = (g_1, \ldots , g_{c - \overline{c}}) is generated by a regular sequence according to Lemma 10.135.6.
\square
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