The Stacks project

80.6 Cycle associated to a coherent sheaf

This is the analogue of Chow Homology, Section 42.10.

Definition 80.6.1. In Situation 80.2.1 let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

  1. For an integral closed subspace $Z \subset X$ with generic point $\xi $ such that $|Z|$ is an irreducible component of $\text{Supp}(\mathcal{F})$ the length of $\mathcal{F}$ at $\xi $ (Definition 80.4.2) is called the multiplicity of $Z$ in $\mathcal{F}$. By Lemma 80.4.4 this is a positive integer.

  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 integral closed subspaces $Z \subset X$ corresponding to irreducible components of $\text{Supp}(\mathcal{F})$ of $\delta $-dimension $k$ and $m_{Z, \mathcal{F}}$ is the multiplicity of $Z$ in $\mathcal{F}$. This is a $k$-cycle by Spaces over Fields, Lemma 70.6.1.

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 80.6.2. In Situation 80.2.1 let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module with $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k$. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. Let $\xi \in |Z|$ be the generic point. Then the coefficient of $Z$ in $[\mathcal{F}]_ k$ is the length of $\mathcal{F}$ at $\xi $.

Proof. Observe that $|Z|$ is an irreducible component of $\text{Supp}(\mathcal{F})$ if and only if $\xi \in \text{Supp}(\mathcal{F})$, see Lemma 80.4.5. Moreover, the length of $\mathcal{F}$ at $\xi $ is zero if $\xi \not\in \text{Supp}(\mathcal{F})$. Combining this with Definition 80.6.1 we conclude. $\square$

Lemma 80.6.3. In Situation 80.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace. If $\dim _\delta (Y) \leq k$, then $[Y]_ k = [i_*\mathcal{O}_ Y]_ k$ where $i : Y \to X$ is the inclusion morphism.

Proof. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. If $Z \not\subset Y$ the $Z$ has coefficient zero in both $[Y]_ k$ and $[i_*\mathcal{O}_ Y]_ k$. If $Z \subset Y$, then the generic point of $Z$ may be viewed as a point $y \in |Y|$ whose image $x \in |X|$. Then the coefficient of $Z$ in $[Y]_ k$ is the length of $\mathcal{O}_ Y$ at $y$ and the coefficient of $Z$ in $[i_*\mathcal{O}_ Y]_ k$ is the length of $i_*\mathcal{O}_ Y$ at $x$. Thus the equality of the coefficients follows from Lemma 80.4.3. $\square$

Lemma 80.6.4. In Situation 80.2.1 let $X/B$ be good. Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of coherent $\mathcal{O}_ X$-modules. Assume that the $\delta $-dimension of the supports of $\mathcal{F}$, $\mathcal{G}$, and $\mathcal{H}$ are $\leq k$. Then $[\mathcal{G}]_ k = [\mathcal{F}]_ k + [\mathcal{H}]_ k$.

Proof. Let $Z$ be an integral closed subspace of $X$ with $\dim _\delta (Z) = k$. It suffices to show that the coefficients of $Z$ in $[\mathcal{G}]_ k$, $[\mathcal{F}]_ k$, and $[\mathcal{H}]_ k$ satisfy the corresponding additivity. By Lemma 80.6.2 it suffices to show

\[ \text{the length of }\mathcal{G}\text{ at }x = \text{the length of }\mathcal{F}\text{ at }x + \text{the length of }\mathcal{H}\text{ at }x \]

for any $x \in |X|$. Looking at Definition 80.4.2 this follows immediately from additivity of lengths, see Algebra, Lemma 10.51.3. $\square$


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