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). $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)