Definition 42.17.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral with $\dim _\delta (X) = n$. Let $f \in R(X)^*$. The principal divisor associated to $f$ is the $(n - 1)$-cycle
\[ \text{div}(f) = \text{div}_ X(f) = \sum \text{ord}_ Z(f) [Z] \]
defined in Divisors, Definition 31.26.5. This makes sense because prime divisors have $\delta $-dimension $n - 1$ by Lemma 42.16.1.
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