Lemma 15.115.5. Let A \to B be an extension of discrete valuation rings. There exists a commutative diagram
\xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u] & A' \ar[u] }
of extensions of discrete valuation rings such that
the extensions K'/K and L'/L of fraction fields are separable algebraic,
the residue fields of A' and B' are separable algebraic closures of the residue fields of A and B, and
if a solution, weak solution, or separable solution exists for A' \to B', then a solution, weak solution, or separable solution exists for A \to B.
Proof.
By Algebra, Lemma 10.159.2 there exists an extension A \subset A' which is a filtered colimit of finite étale extensions such that the residue field of A' is a separable algebraic closure of the residue field of A. Then A \subset A' is an extension of discrete valuation rings such that the induced extension K'/K of fraction fields is separable algebraic.
Let B \subset B' be a strict henselization of B. Then B \subset B' is an extension of discrete valuation rings whose fraction field extension is separable algebraic. By Algebra, Lemma 10.155.9 there exists a commutative diagram as in the statement of the lemma. Parts (1) and (2) of the lemma are clear.
Let K'_1/K' be a (weak) solution for A' \to B'. Since A' is a colimit, we can find a finite étale extension A \subset A_1' and a finite extension K_1 of the fraction field F of A_1' such that K'_1 = K' \otimes _ F K_1. As A \subset A_1' is finite étale and B' strictly henselian, it follows that B' \otimes _ A A_1' is a finite product of rings isomorphic to B'. Hence
L' \otimes _ K K_1 = L' \otimes _ K F \otimes _ F K_1
is a finite product of rings isomorphic to L' \otimes _{K'} K'_1. Thus we see that K_1/K is a (weak) solution for A \to B'. Hence it is also a (weak) solution for A \to B by Lemma 15.115.4.
\square
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