The Stacks project

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.

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