Lemma 34.10.6. Let $T$ be an affine scheme. Let $\{ T_ j \to T\} _{j = 1, \ldots , m}$ be a family of morphisms with $T_ j$ affine for all $j$. The following are equivalent
$\{ T_ j \to T\} _{j = 1, \ldots , m}$ is a standard V covering,
there is a standard V covering which refines $\{ T_ j \to T\} _{j = 1, \ldots , m}$, and
$\{ \coprod _{j = 1, \ldots , m} T_ j \to T\} $ is a standard V covering.
Proof. Omitted. Hints: This follows almost immediately from the definition. The only slightly interesting point is that a morphism from the spectrum of a local ring into $\coprod _{j = 1, \ldots , m} T_ j$ must factor through some $T_ j$. $\square$
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