Remark 67.19.3 (Variant for complete discrete valuation rings). In Lemmas 67.19.1 and 67.19.2 it suffices to consider complete discrete valuation rings. To be precise in Lemma 67.19.1 we can replace condition (3) by the following condition: Given any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] \ar@{-->}[ru] & Y }$

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists at most one dotted arrow making the diagram commute. Namely, given any diagram as in Lemma 67.19.1 (3) the completion $A^\wedge$ is a discrete valuation ring (More on Algebra, Lemma 15.43.5) and the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A^\wedge ) \to X$ implies the uniqueness of the arrow $\mathop{\mathrm{Spec}}(A) \to X$ for example by Properties of Spaces, Proposition 64.17.1. Similarly in Lemma 67.19.2 we can replace condition (3) by the following condition: Given any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

where $A$ is a complete discrete valuation ring with fraction field $K$ there exists an extension $A \subset A'$ of complete discrete valuation rings inducing a fraction field extension $K \subset K'$ such that there exists a unique arrow $\mathop{\mathrm{Spec}}(A') \to X$ making the diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(K') \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A') \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

commute. Namely, given any diagram as in Lemma 67.19.2 part (3) the existence of any commutative diagram

$\xymatrix{ \mathop{\mathrm{Spec}}(L) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K) \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(B) \ar[r] \ar[rru] & \mathop{\mathrm{Spec}}(A) \ar[r] & Y }$

for any extension $A \subset B$ of discrete valuation rings will imply there exists an arrow $\mathop{\mathrm{Spec}}(A) \to X$ fitting into the diagram. This was shown in Morphisms of Spaces, Lemma 65.41.4. In fact, it follows from these considerations that it suffices to look for dotted arrows in diagrams for any class of discrete valuation rings such that, given any discrete valuation ring, there is an extension of it that is in the class. For example, we could take complete discrete valuation rings with algebraically closed residue field.

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