I am having trouble with computing the intersection number using the following definition [Ref Vakil Chapter 20]:
Suppose $\mathcal{F}$ is a coherent sheaf on $X$ with proper support of dimension at most $n$, and $\mathcal{L}_{1}$,..., $\mathcal{L}_{n}$ are invertible sheaves on $X$. The intersection of the line bundles with the coherent sheaf is defined to be $$(\mathcal{L}_1\cdot...\mathcal{L}_n\cdot\mathcal{F})=\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n\}}(-1)^m\chi(X,\mathcal{L}_{i_1}^\vee\otimes...\otimes\mathcal{L}_{i_m}^\vee\otimes\mathcal{F}).$$
So trying to attempt both questions 20.1.B(b) and 20.1.C, I run unto certain problems understanding the definition while doing the calculations:
Exercise 20.1.B(b) [Proving Bézout's theorem] Suppose $k$ is an infinite field, $X=\mathbb{P}^N$ and $Y$ is a dimension $n$ subvariety of $X$. If $H_1$, ..., $H_{n}$ are generally chosen hypersurfaces of degree $d_1$, ..., $d_n$ respectively, show that
$$(\mathcal{O}_X(H_1)\cdot...\cdot\mathcal{O}_X(H_n)\cdot Y)=d_{1}...d_{n}\deg Y$$
Suppose exercise 20.1.C is true that is
Exercise 20.1.C Suppose $D$ is an effective Cartier divisor on $X$ that restricts to an effective Cartier divisor $D|_Y$ on $Y$, show that $$(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot \mathcal{O}(D)\cdot Y)=(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot D|_Y).$$ More generally if $D$ is an effective Cartier divisor on $X$ that does not contain any associated point of $\mathcal{F}$, show that $$(\mathcal{L}_1\cdot...\cdot \mathcal{L}_n-1\cdot \mathcal{O}(D)\cdot \mathcal{F})=(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot \mathcal{F}|_Y).$$
Suppose exercise C is true, then we may prove by induction I think, on the dimension $n$ of $Y$. Suppose it is true for $n-1$, then the formula in C gives
$$(\mathcal{O}_X(H_1)\cdot...\cdot\mathcal{O}_X(H_n)\cdot Y)=(\mathcal{O}_Y(H_1)\cdot...\cdot\mathcal{O}_Y(H_{n-1})\cdot\mathcal{O}_Y(H_n))\\ =\deg_{Y}(H_1)\cdot...\cdot\deg_{Y}(H_n) (\text{Or } =\deg_{Y}(H_{1}\cap...H_{n})???.)$$
Then we conclude by $\deg_{Y}K=\deg K\cdot \deg Y$ (?????). Then I will have a problem, either I get $d_{1}...d_{n}(\deg Y)^n$ or $d_{1}...d_{n}(\deg Y)$. Which one is correct?
Assume for the sake of consistency that the second one is correct...
Which I think remains to solve C. I am stuck at this stage...
So using the following exact sequence $$0\rightarrow \mathcal{O}_X(-Y)\rightarrow\mathcal{O}_X\rightarrow \mathcal{O}_X|_{Y}\rightarrow 0,$$
by tensoring with the any line bundle $\mathcal{L}$ we have $$\chi(X,\mathcal{L}^\vee\otimes\mathcal{O}_{X}(-Y))=\chi(X,\mathcal{L}^\vee)-\chi(Y,\mathcal{L}^\vee|_Y).$$
Hence doing this for several line bundles and taking signed sum, we get
$$(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n}\cdot Y)=\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n\}}(-1)^m\chi(X,\mathcal{L}_{i_1}^\vee\otimes...\otimes\mathcal{L}_{i_m}^\vee)+\sum_{\{i_{1},...,i_{m}\}\subseteq \{1,...,n\}}(-1)^{m-1}\chi(Y,\mathcal{L}_{i_1}|_{Y}^\vee\otimes...\otimes\mathcal{L}_{i_m}|_{Y}^\vee),$$
where it is suggested that the first term in the RHS should vanish and the second term should be something like $(\mathcal{L}_1\cdot...\cdot \mathcal{L}_{n-1}\cdot D|_Y)$, but I don't see why is this evident. I am quite lost in all these manipulations.