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
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
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.
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Your email address will not be published.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)