The Stacks project

Lemma 115.23.10. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $\{ D_ i\} _{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given $n_ i \geq 0$ for $i \in I$. Then

\[ [D]_{n - 1} = \sum \nolimits _ i n_ i[D_ i]_{n - 1} \]

in $Z_{n - 1}(X)$.

Proof. Since we are proving an equality of cycles we may work locally on $X$. Hence this reduces to a finite sum, and by induction to a sum of two effective Cartier divisors $D = D_1 + D_2$. By Chow Homology, Lemma 42.24.2 we see that $D_1 = \text{div}_{\mathcal{O}_ X(D_1)}(1_{D_1})$ where $1_{D_1}$ denotes the canonical section of $\mathcal{O}_ X(D_1)$. Of course we have the same statement for $D_2$ and $D$. Since $1_ D = 1_{D_1} \otimes 1_{D_2}$ via the identification $\mathcal{O}_ X(D) = \mathcal{O}_ X(D_1) \otimes \mathcal{O}_ X(D_2)$ we win by Divisors, Lemma 31.27.5. $\square$

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