What conditions are needed for intersection number of Cartier divisors to equal dimension of global sections?

Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed). Let $F$ be a coherent sheaf with proper support (not sure what this means exactly. I think it means the morphism $\operatorname{supp}(F) \rightarrow X \rightarrow k$ where $\operatorname{supp}(F)$ has the reduced structure, is a proper morphism) and suppose $\operatorname{supp}(F)$ has dimension at most $n$. Let $L_1, \dotsc, L_n$ be invertible sheaves on $X$. Define the intersection multiplicity $$\operatorname{int}(L_1,\dotsc, L_n, F) := \sum_{S \subseteq \{1, \dotsc,n\}} (-1)^{|S|} \chi\bigl(F \otimes (\textstyle\bigotimes_{i \in S} L_i^\vee)\bigr),$$ where $\chi$ is Euler characteristic (alternating sum of dimensions of cohomology groups).

Question: Suppose $L_i = \mathcal{O}(D_i)$ for some effective Cartier divisor $D_i$ on $X$. Let $Y = \bigcap_{i=1,\dotsc,n} D_i$ be the scheme-theoretic intersection of the $D_i$ and $i : Y \rightarrow X$ be the inclusion. Suppose furthermore that $\dim(Y) = 0$. Do we have $$\dim_k \Gamma(Y, i^* F) = \operatorname{int}(L_1, \dotsc , L_n, F)\,?$$ Here, $\dim_k$ is dimension as a vector space, not the dimension of $Y$. If not, what conditions are required for it to be true? In particular, is this true if for every $y \in Y = \bigcap_{i=1,\dotsc,n} D_i$, the defining equation of $D_i$ at $i(y) \in X$ forms a regular sequence of $\mathcal{O}_{X, i(y)}$ and $F_{i(y)}$?

There is Vakil 20.1.D: if $j : D \rightarrow X$ is an effective Cartier divisors that doesn't contain any associated point of $F$, then $$\operatorname{int}_X(L_1, \dotsc, L_n = \mathcal{O}(D), F) = \operatorname{int}_D(j^* L_1 , \dotsc, j^* L_{n-1}, j^* F).$$ So, if every $L_i = \mathcal{O}(D_i)$ then I think we can get it to be true. However, I'm not sure if $j^* \mathcal{O}(D_{i-1})$ is still the invertible sheaf associated to an effective Cartier divisor on $D_n$, and even if it is, if pulling back to $D_n$ preserves the property of not containing any associated point.