The Stacks project

Lemma 81.10.3. In Situation 81.2.1 let $f : X \to Y$ be an ├ętale morphism of good algebraic spaces over $B$. If $Z \subset Y$ is an integral closed subspace, then $f^*[Z] = \sum [Z']$ where the sum is over the irreducible components (Remark 81.5.1) of $f^{-1}(Z)$.

Proof. The meaning of the lemma is that the coefficient of $[Z']$ is $1$. This follows from the fact that $f^{-1}(Z)$ is a reduced algebraic space because it is ├ętale over the integral algebraic space $Z$. $\square$

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