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The Stacks project

Lemma 61.10.1. Let Y be an affine scheme. Let X = \mathop{\mathrm{lim}}\nolimits X_ i be a directed limit of affine schemes over Y. The following are equivalent

  1. \{ X \to Y\} is a standard V covering (Topologies, Definition 34.10.1), and

  2. \{ X_ i \to Y\} is a standard V covering for all i.

Proof. A singleton \{ X \to Y\} is a standard V covering if and only if given a morphism g : \mathop{\mathrm{Spec}}(V) \to Y there is an extension of valuation rings V \subset W and a commutative diagram

\xymatrix{ \mathop{\mathrm{Spec}}(W) \ar[r] \ar[d] & X \ar[d] \\ \mathop{\mathrm{Spec}}(V) \ar[r]^ g & Y }

Thus (1) \Rightarrow (2) is immediate from the definition. Conversely, assume (2) and let g : \mathop{\mathrm{Spec}}(V) \to Y as above be given. Write \mathop{\mathrm{Spec}}(V) \times _ Y X_ i = \mathop{\mathrm{Spec}}(A_ i). Since \{ X_ i \to Y\} is a standard V covering, we may choose a valuation ring W_ i and a ring map A_ i \to W_ i such that the composition V \to A_ i \to W_ i is an extension of valuation rings. In particular, the quotient A'_ i of A_ i by its V-torsion is a faitfhully flat V-algebra. Flatness by More on Algebra, Lemma 15.22.10 and surjectivity on spectra because A_ i \to W_ i factors through A'_ i. Thus

A = \mathop{\mathrm{colim}}\nolimits A'_ i

is a faithfully flat V-algebra (Algebra, Lemma 10.39.20). Since \{ \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(V)\} is a standard fpqc cover, it is a standard V cover (Topologies, Lemma 34.10.2) and hence we can choose \mathop{\mathrm{Spec}}(W) \to \mathop{\mathrm{Spec}}(A) such that V \to W is an extension of valuation rings. Since we can compose with the morphism \mathop{\mathrm{Spec}}(A) \to X = \mathop{\mathrm{Spec}}(\mathop{\mathrm{colim}}\nolimits A_ i) the proof is complete. \square


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