The Stacks project

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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