The Stacks project

43.3 Cycles

Let $X$ be a variety. A closed subvariety of $X$ is an integral closed subscheme $Z \subset X$. A $k$-cycle on $X$ is a finite formal sum $\sum n_ i [Z_ i]$ where each $Z_ i$ is a closed subvariety of dimension $k$. Whenever we use the notation $\alpha = \sum n_ i[Z_ i]$ for a $k$-cycle we always assume the subvarieties $Z_ i$ are pairwise distinct and $n_ i \not= 0$ for all $i$. In this case the support of $\alpha $ is the closed subset

\[ \text{Supp}(\alpha ) = \bigcup Z_ i \subset X \]

of dimension $k$. The group of $k$-cycles is denoted $Z_ k(X)$. See Chow Homology, Section 42.8.


Comments (0)


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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AZ9. Beware of the difference between the letter 'O' and the digit '0'.