The Stacks project

Remark 42.29.3. Let $X$ be a scheme locally of finite type over $S$ as in Situation 42.7.1. In [F] a pseudo-divisor on $X$ is defined as a triple $D = (\mathcal{L}, Z, s)$ where $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, $Z \subset X$ is a closed subset, and $s \in \Gamma (X \setminus Z, \mathcal{L})$ is a nowhere vanishing section. Similarly to the above, one can define for every $\alpha $ in $\mathop{\mathrm{CH}}\nolimits _{k + 1}(X)$ a product $D \cdot \alpha $ in $\mathop{\mathrm{CH}}\nolimits _ k(Z \cap |\alpha |)$ where $|\alpha |$ is the support of $\alpha $.