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The Stacks project

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

  1. \{ T_ j \to T\} _{j = 1, \ldots , m} is a standard V covering,

  2. there is a standard V covering which refines \{ T_ j \to T\} _{j = 1, \ldots , m}, and

  3. \{ \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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