Lemma 34.5.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be a smooth covering of $T$. Then there exists a smooth 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 an affine scheme, and such that each morphism $U_ j \to T$ is standard smooth, see Morphisms, Definition 29.34.1. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$.

Proof. Omitted, but see Algebra, Lemma 10.137.10. $\square$

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