## 42.10 Cycle associated to a coherent sheaf

Lemma 42.10.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

1. The collection of irreducible components of the support of $\mathcal{F}$ is locally finite.

2. Let $Z' \subset \text{Supp}(\mathcal{F})$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then

$\text{length}_{\mathcal{O}_{X, \xi }} \mathcal{F}_\xi < \infty$
3. If $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$ and $\xi \in \text{Supp}(\mathcal{F})$ with $\delta (\xi ) = k$, then $\xi$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F})$.

Proof. By Cohomology of Schemes, Lemma 30.9.7 the support $Z$ of $\mathcal{F}$ is a closed subset of $X$. We may think of $Z$ as a reduced closed subscheme of $X$ (Schemes, Lemma 26.12.4). Hence (1) follows from Divisors, Lemma 31.26.1 applied to $Z$ and (3) follows from Lemma 42.9.1 applied to $Z$.

Let $\xi \in Z'$ be as in (2). In this case for any specialization $\xi ' \leadsto \xi$ in $X$ we have $\mathcal{F}_{\xi '} = 0$. Recall that the non-maximal primes of $\mathcal{O}_{X, \xi }$ correspond to the points of $X$ specializing to $\xi$ (Schemes, Lemma 26.13.2). Hence $\mathcal{F}_\xi$ is a finite $\mathcal{O}_{X, \xi }$-module whose support is $\{ \mathfrak m_\xi \}$. Hence it has finite length by Algebra, Lemma 10.62.3. $\square$

Definition 42.10.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

1. For any irreducible component $Z' \subset \text{Supp}(\mathcal{F})$ with generic point $\xi$ the integer $m_{Z', \mathcal{F}} = \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{F}_\xi$ (Lemma 42.10.1) is called the multiplicity of $Z'$ in $\mathcal{F}$.

2. Assume $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$. The $k$-cycle associated to $\mathcal{F}$ is

$[\mathcal{F}]_ k = \sum m_{Z', \mathcal{F}}[Z']$

where the sum is over the irreducible components of $\text{Supp}(\mathcal{F})$ of $\delta$-dimension $k$. (This is a $k$-cycle by Lemma 42.10.1.)

It is important to note that we only define $[\mathcal{F}]_ k$ if $\mathcal{F}$ is coherent and the $\delta$-dimension of $\text{Supp}(\mathcal{F})$ does not exceed $k$. In other words, by convention, if we write $[\mathcal{F}]_ k$ then this implies that $\mathcal{F}$ is coherent on $X$ and $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$.

Lemma 42.10.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme. If $\dim _\delta (Z) \leq k$, then $[Z]_ k = [{\mathcal O}_ Z]_ k$.

Proof. This is because in this case the multiplicities $m_{Z', Z}$ and $m_{Z', \mathcal{O}_ Z}$ agree by definition. $\square$

Lemma 42.10.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent sheaves on $X$. Assume that the $\delta$-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ is $\leq k$. Then $[\mathcal{G}]_ k = [\mathcal{F}]_ k + [\mathcal{H}]_ k$.

Proof. Follows immediately from additivity of lengths, see Algebra, Lemma 10.52.3. $\square$

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