Lemma 34.3.3 (020Q)—The Stacks project

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Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.3: The Zariski topology
Lemma 34.3.3 (cite)

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Lemma 34.3.3. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a Zariski covering of $T$ . Then there exists a Zariski covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is a standard open of $T$ , see Schemes, Definition 26.5.2. Moreover, we may choose each $U_ j$ to be an open of one of the $T_ i$ .

Proof. Follows as $T$ is quasi-compact and standard opens form a basis for its topology. This is also proved in Schemes, Lemma 26.5.1. $\square$

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