Lemma 61.10.2. Let $X \to Y$ be a morphism of affine schemes. The following are equivalent

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

2. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\}$ is a ph covering for each $i$, and

3. $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of affine schemes over $Y$ such that $\{ X_ i \to Y\}$ is an h covering for each $i$.

Proof. Proof of (2) $\Rightarrow$ (1). Recall that a V covering given by a single arrow between affines is a standard V covering, see Topologies, Definition 34.10.7 and Lemma 34.10.6. Recall that any ph covering is a V covering, see Topologies, Lemma 34.10.10. Hence if $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as in (2), then $\{ X_ i \to Y\}$ is a standard V covering for each $i$. Thus by Lemma 61.10.1 we see that (1) is true.

Proof of (3) $\Rightarrow$ (2). This is clear because an h covering is always a ph covering, see More on Flatness, Definition 38.34.2.

Proof of (1) $\Rightarrow$ (3). This is the interesting direction, but the interesting content in this proof is hidden in More on Flatness, Lemma 38.34.1. Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(R)$. We can write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ with $A_ i$ of finite presentation over $R$, see Algebra, Lemma 10.127.2. Set $X_ i = \mathop{\mathrm{Spec}}(A_ i)$. Then $\{ X_ i \to Y\}$ is a standard V covering for all $i$ by (1) and Topologies, Lemma 34.10.6. Hence $\{ X_ i \to Y\}$ is an h covering by More on Flatness, Definition 38.34.2. This finishes the proof. $\square$

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