The Stacks project

Lemma 42.69.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{K}_ X(\mathcal{L}))$ be a meromorphic section of $\mathcal{L}$. Assume

  1. $\dim _\delta (X) \leq k + 1$,

  2. $X$ has no embedded points,

  3. $\mathcal{F}$ has no embedded associated points,

  4. the support of $\mathcal{F}$ is $X$, and

  5. the section $s$ is regular meromorphic.

In this situation let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal of denominators of $s$, see Divisors, Definition 31.23.10. Then we have the following:

  1. there are short exact sequences

    \[ \begin{matrix} 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow {1} & \mathcal{F} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{I}\mathcal{F} & \xrightarrow {s} & \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix} \]
  2. the coherent sheaves $\mathcal{Q}_1$, $\mathcal{Q}_2$ are supported in $\delta $-dimension $\leq k$,

  3. the section $s$ restricts to a regular meromorphic section $s_ i$ on every irreducible component $X_ i$ of $X$ of $\delta $-dimension $k + 1$, and

  4. writing $[\mathcal{F}]_{k + 1} = \sum m_ i[X_ i]$ we have

    \[ [\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = \sum m_ i(X_ i \to X)_*\text{div}_{\mathcal{L}|_{X_ i}}(s_ i) \]

    in $Z_ k(X)$, in particular

    \[ [\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1} \]

    in $\mathop{\mathrm{CH}}\nolimits _ k(X)$.

Proof. Recall from Divisors, Lemma 31.24.5 the existence of injective maps $1 : \mathcal{I}\mathcal{F} \to \mathcal{F}$ and $s : \mathcal{I}\mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X}\mathcal{L}$ whose cokernels are supported on a closed nowhere dense subsets $T$. Denote $\mathcal{Q}_ i$ there cokernels as in the lemma. We conclude that $\dim _\delta (\text{Supp}(\mathcal{Q}_ i)) \leq k$. By Divisors, Lemmas 31.23.5 and 31.23.8 the pullbacks $s_ i$ are defined and are regular meromorphic sections for $\mathcal{L}|_{X_ i}$. The equality of cycles in (4) implies the equality of cycle classes in (4). Hence the only remaining thing to show is that

\[ [\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k = \sum m_ i(X_ i \to X)_*\text{div}_{\mathcal{L}|_{X_ i}}(s_ i) \]

holds in $Z_ k(X)$. To see this, let $Z \subset X$ be an integral closed subscheme of $\delta $-dimension $k$. Let $\xi \in Z$ be the generic point. Let $A = \mathcal{O}_{X, \xi }$ and $M = \mathcal{F}_\xi $. Moreover, choose a generator $s_\xi \in \mathcal{L}_\xi $. Then we can write $s = (a/b) s_\xi $ where $a, b \in A$ are nonzerodivisors. In this case $I = \mathcal{I}_\xi = \{ x \in A \mid x(a/b) \in A\} $. In this case the coefficient of $[Z]$ in the left hand side is

\[ \text{length}_ A(M/(a/b)IM) - \text{length}_ A(M/IM) \]

and the coefficient of $[Z]$ in the right hand side is

\[ \sum \text{length}_{A_{\mathfrak q_ i}}(M_{\mathfrak q_ i}) \text{ord}_{A/\mathfrak q_ i}(a/b) \]

where $\mathfrak q_1, \ldots , \mathfrak q_ t$ are the minimal primes of the $1$-dimensional local ring $A$. Hence the result follows from Lemma 42.69.5. $\square$

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