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.

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