## 43.5 Cycle associated to a coherent sheaf

Suppose that $X$ is a variety and that $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module with $\dim (\text{Supp}(\mathcal{F})) \leq k$. Let $Z_ i$ be the irreducible components of $\text{Supp}(\mathcal{F})$ of dimension $k$ and let $n_ i$ be the multiplicity of $Z_ i$ in $\mathcal{F}$ defined as

$n_ i = \text{length}_{\mathcal{O}_{X, Z_ i}} \mathcal{F}_{\xi _ i}$

where $\mathcal{O}_{X, Z_ i}$ is the local ring of $X$ at the generic point $\xi _ i$ of $Z_ i$ and $\mathcal{F}_{\xi _ i}$ is the stalk of $\mathcal{F}$ at this point. We define the $k$-cycle associated to $\mathcal{F}$ to be the $k$-cycle

$[\mathcal{F}]_ k = \sum n_ i [Z_ i].$

See Chow Homology, Section 42.10. Note that, if $Z \subset X$ is a closed subscheme with $\dim (Z) \leq k$, then $[Z]_ k = [\mathcal{O}_ Z]_ k$ by definition.

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.

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 0AZB. Beware of the difference between the letter 'O' and the digit '0'.