Lemma 62.7.4. 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. Then to check that \alpha is equidimensional we may work Zariski locally on X and S.
Proof. Namely, the condition that \alpha is equidimensional just means that the closure of the support of \alpha has relative dimension \leq r over S. Since taking closures commutes with restriction to opens, the lemma follows (small detail omitted). \square
Comments (0)