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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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