Definition 59.15.1. Let $T$ be a scheme. An *fpqc covering* of $T$ is a family $\{ \varphi _ i : T_ i \to T\} _{i \in I}$ such that

each $\varphi _ i$ is a flat morphism and $\bigcup _{i\in I} \varphi _ i(T_ i) = T$, and

for each affine open $U \subset T$ there exists a finite set $K$, a map $\mathbf{i} : K \to I$ and affine opens $U_{\mathbf{i}(k)} \subset T_{\mathbf{i}(k)}$ such that $U = \bigcup _{k \in K} \varphi _{\mathbf{i}(k)}(U_{\mathbf{i}(k)})$.

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