Lemma 62.7.5. Let f : X \to S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r \geq 0 be an integer. Let \alpha be a relative r-cycle on X/S. Let \{ g_ i : S_ i \to S\} be an fppf covering. Then \alpha is equidimensional if and only if each base change g_ i^*\alpha is equidimensional.
Proof. If \alpha is equidimensional, then each g_ i^*\alpha is too by Lemma 62.7.3. Assume each g_ i^*\alpha is equidimensional. Denote W the closure of \text{Supp}(\alpha ) in X. Since g_ i : S_ i \to S is universally open (being flat and locally of finite presentation), so is the morphism f_ i : X_ i = S_ i \times _ S X \to X. Denote \alpha _ i = g_ i^*\alpha . We have \text{Supp}(\alpha _ i) = f_ i^{-1}(\text{Supp}(\alpha )) by Lemma 62.5.7. Since f_ i is open, we see that W_ i = f_ i^{-1}(W) is the closure of \text{Supp}(\alpha _ i). Hence by assumption the morphism W_ i \to S_ i has relative dimension \leq r. By Morphisms, Lemma 29.28.3 (and the fact that the morphisms S_ i \to S are jointly surjective) we conclude that W \to S has relative dimension \leq r. \square
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