Support of a Weil divisor equals to support of Cartier divisor?

Question

Sorry for my bad English.

Let $X$ be locally fractional integral Noetherian scheme on which we can identify Weil divisor and Cartier divisor (cf. Hartshorne II.Prop.6.11).

If $D$ is a Cartier divisor, we define support of $D$ as

Supp$D:=\{x\in X| D_x\neq 1\}$

where $D_x$ is stalk at $x$ of $D\in {\scr K}^*/{\scr O}^*_X$.

On the other hand, we can think $D$ as a Weil divisor, so $D=\sum_{i} n_i Y_i$ where $Y_i$ are prime divisors and $n_i\in \mathbb{Z}\backslash \{0\}$.

Now Supp$D=\bigcup_i Y_i$?

Please help me, thanks.

Answer 1

The answer is positive. It suffices to first consider the case when $D\geq 0$, which follows from [EGAIV$_4$, 21.6.6.2] that $\mathrm{Supp}(\mathrm{cyc}(D))=\mathrm{Supp}(D)$ and the definition [EGAIV$_4$, 21.6.1]. Here $\mathrm{cyc}: \mathrm{Div}^+(X)\rightarrow \mathfrak{Z}^1(X)$ is the homomorphism from the monoid of effective Cartier divisors to the group of codimension-one cycles (Weil divisors).