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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