16.14 Approximation for henselian pairs
We can generalize the discussion of Section 16.13 to the case of henselian pairs. Henselian pairs where defined in More on Algebra, Section 15.11.
Lemma 16.14.1.slogan Let (A, I) be a henselian pair with A Noetherian. Let A^\wedge be the I-adic completion of A. Assume at least one of the following conditions holds
A \to A^\wedge is a regular ring map,
A is a Noetherian G-ring, or
(A, I) is the henselization (More on Algebra, Lemma 15.12.1) of a pair (B, J) where B is a Noetherian G-ring.
Given f_1, \ldots , f_ m \in A[x_1, \ldots , x_ n] and \hat{a}_1, \ldots , \hat{a}_ n \in A^\wedge such that f_ j(\hat{a}_1, \ldots , \hat{a}_ n) = 0 for j = 1, \ldots , m, for every N \geq 1 there exist a_1, \ldots , a_ n \in A such that \hat{a}_ i - a_ i \in I^ N and such that f_ j(a_1, \ldots , a_ n) = 0 for j = 1, \ldots , m.
Proof.
By More on Algebra, Lemma 15.50.15 we see that (3) implies (2). By More on Algebra, Lemma 15.50.14 we see that (2) implies (1). Thus it suffices to prove the lemma in case A \to A^\wedge is a regular ring map.
Let \hat{a}_1, \ldots , \hat{a}_ n be as in the statement of the lemma. By Theorem 16.12.1 we can find a factorization A \to B \to A^\wedge with A \to P smooth and b_1, \ldots , b_ n \in B with f_ j(b_1, \ldots , b_ n) = 0 in B. Denote \sigma : B \to A^\wedge \to A/I^ N the composition. By More on Algebra, Lemma 15.9.14 we can find an étale ring map A \to A' which induces an isomorphism A/I^ N \to A'/I^ NA' and an A-algebra map \tilde\sigma : B \to A' lifting \sigma . Since (A, I) is henselian, there is an A-algebra map \chi : A' \to A, see More on Algebra, Lemma 15.11.6. Then setting a_ i = \chi (\tilde\sigma (b_ i)) gives a solution.
\square
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