Remark 42.29.2. Let $X$ be a scheme locally of finite type over $S$ as in Situation 42.7.1. Let $(D, \mathcal{N}, \sigma )$ be a triple consisting of a locally principal (Divisors, Definition 31.13.1) closed subscheme $i : D \to X$, an invertible $\mathcal{O}_ D$-module $\mathcal{N}$, and a surjection $\sigma : \mathcal{N}^{\otimes -1} \to i^*\mathcal{I}_ D$ of $\mathcal{O}_ D$-modules^{1}. Here $\mathcal{N}$ should be thought of as a *virtual normal bundle of $D$ in $X$*. The construction of $i^* : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D)$ in Definition 42.29.1 generalizes to such triples, see Section 42.54.

[1] This condition assures us that if $D$ is an effective Cartier divisor, then $\mathcal{N} = \mathcal{O}_ X(D)|_ D$.

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