Lemma 76.19.7. Let S be a scheme. Consider a commutative diagram
\xymatrix{ X \ar[d] & T \ar[l] \ar[d] \\ Y & T' \ar[l] }
of algebraic spaces over S where X \to Y is smooth and T \to T' is a thickening. Then there exists an étale covering \{ T'_ i \to T'\} such that we can find the dotted arrow in
\xymatrix{ X \ar[d] & T \ar[l] \ar[d] & T \times _{T'} T'_ i \ar[l] \ar[d] \\ Y & T' \ar[l] & T'_ i \ar[l] \ar@{..>}[llu] }
making the diagram commute (for all i).
Proof.
Choose an étale covering \{ Y_ i \to Y\} with each Y_ i affine. After replacing T' by the induced étale covering we may assume Y is affine.
Assume Y is affine. Choose an étale covering \{ X_ i \to X\} . This gives rise to an étale covering of T. This étale covering of T comes from an étale covering of T' (by Theorem 76.8.1, see discussion in Section 76.9). Hence we may assume X is affine.
Assume X and Y are affine. We can do one more étale covering of T' and assume T' is affine. In this case the lemma follows from Algebra, Lemma 10.138.17.
\square
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