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The Stacks project

Lemma 62.7.6. Let f : X \to S be a morphism of schemes. Assume S locally Noetherian and f locally of finite type. Let r, e \geq 0 be integers. Let \alpha be a relative r-cycle on X/S. Let \{ f_ i : X_ i \to X\} be a jointly surjective family of flat morphisms, locally of finite type, and of relative dimension e. Then \alpha is equidimensional if and only if each flat pullback f_ i^*\alpha is equidimensional.

Proof. Omitted. Hint: As in the proof of Lemma 62.7.5 one shows that the inverse image by f_ i of the closure W of the support of \alpha is the closure W_ i of the support of f_ i^*\alpha . Then W \to S has relative dimension \leq r holds if W_ i \to S has relative dimension \leq r + e for all i. \square


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