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43.4 Cycle associated to closed subscheme

Suppose that $X$ is a variety and that $Z \subset X$ be a closed subscheme with $\dim (Z) \leq k$. Let $Z_ i$ be the irreducible components of $Z$ of dimension $k$ and let $n_ i$ be the multiplicity of $Z_ i$ in $Z$ defined as

\[ n_ i = \text{length}_{\mathcal{O}_{X, Z_ i}} \mathcal{O}_{Z, Z_ i} \]

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

\[ [Z]_ k = \sum n_ i [Z_ i]. \]

See Chow Homology, Section 42.9.

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