Example 58.15.3. Examples of fpqc coverings.

1. Any Zariski open covering of $T$ is an fpqc covering.

2. A family $\{ \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)\}$ is an fpqc covering if and only if $A \to B$ is a faithfully flat ring map.

3. If $f: X \to Y$ is flat, surjective and quasi-compact, then $\{ f: X\to Y\}$ is an fpqc covering.

4. The morphism $\varphi : \coprod _{x \in \mathbf{A}^1_ k} \mathop{\mathrm{Spec}}(\mathcal{O}_{\mathbf{A}^1_ k, x}) \to \mathbf{A}^1_ k$, where $k$ is a field, is flat and surjective. It is not quasi-compact, and in fact the family $\{ \varphi \}$ is not an fpqc covering.

5. Write $\mathbf{A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$. Denote $i_ x : D(x) \to \mathbf{A}^2_ k$ and $i_ y : D(y) \hookrightarrow \mathbf{A}^2_ k$ the standard opens. Then the families $\{ i_ x, i_ y, \mathop{\mathrm{Spec}}(k[[x, y]]) \to \mathbf{A}^2_ k\}$ and $\{ i_ x, i_ y, \mathop{\mathrm{Spec}}(\mathcal{O}_{\mathbf{A}^2_ k, 0}) \to \mathbf{A}^2_ k\}$ are fpqc coverings.

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