I want to ask whether the proof of (2.4) in Fulton can be simplified under better assumptions. Instead of copying all the definitions and constructions from that specific book, let me state things a bit more vaguely and generally.
Let $X$ be a variety, and let $D$ and $D'$ be two Cartier divisors. It is often said that defining the intersection product is subtle when $D$ and $D'$ do not intersect properly; that is, $\text{supp}(D)\cap\text{supp}(D')$ contains a prime Weil divisor. One way to address this is through moving lemma. Alternatively, Fulton just defines the intersection by pulling back $\mathcal{O}_D$ to the associated Weil divisor $[D']$ (linearly if $[D']=\sum n_iV_i$), but then he has to prove the commutativity (2.4): pulling back $D$ to $[D']$ equals to pulling back $D'$ to $[D]$.
The proof requires blowup to (roughly saying) express the common part of $D$ and $D'$ by same divisor. So it is natural to ask:
Question: if $X$ is normal, so each Weil divisor is associated with at most one Cartier divisor, then do we no longer need the moving lemma or blowing-up techniques to define the class $D\cdot D'$ in Chow group?
My sketch (for 2.4): for simplicity assume that $V$ is the only common codimension-1 component of $\text{supp}(D)\cap\text{supp}(D')$. Since $X$ is normal, we can write $D = nV + C$, $D'=mV + C'$ where $C$ intersects $D'$ properly and vice versa. Then the problem reduces to proving that $nV\cdot [mV] = mV\cdot [nV]$ (in Fulton's definitions/notations), which is equivalent to $(\mathcal{O}(nV)|_V)^{\otimes m}\simeq(\mathcal{O}(mV)|_V)^{\otimes n}$.
Some remarks: When $X$ is sufficiently nice, morally it should corresponds to $[\mathcal{O}_D]\cdot[\mathcal{O}_{D'}]$ in $K_0(X)$. This approach should be equivalent via Chern character, also hopefully without denominators, since we are only working with divisors. Commutativity is obvious here and calculations involve something like balancing Tor, but I cannot rigorously prove the equivalence (at least I think we need GRR for $D\subset X$, which is a nontrivial theorem). I would appreciate it if someone could check the specific proof of (2.4) under the assumption that $X$ is normal.
