# Intersection Number vs Intersection Multiplicity

I am studying from https://userpage.fu-berlin.de/aconstant/Alg2/Bib/Shafarevich.pdf.

Let $$X$$ be a smooth irreducible quasi-projective variety of dimension $$n$$ (over an algebraically closed field $$k$$) and $$D_1,\ldots,D_j$$ be effective divisors on $$X$$ in general position (where $$j\le n$$). Shafarevich defines two different notions (although one is only defined for $$j=n$$, but has an immediate and obvious generalization): intersection multiplicity and intersection number of $$D_1,\ldots,D_j$$.

In both cases, we consider the irreducible components $$C_1,\ldots,C_r$$ of $$\bigcap_{i=1}^j\text{Supp}(D_i)$$ and for a particular $$C_i$$, we choose $$x\in C_i$$ and local equations $$f_1,\ldots,f_j$$ of $$D_1,\ldots,D_j$$ respectively at $$x$$. Then we consider the set $$\mathcal O_{X,{C_i}}/(f_1,\ldots,f_j)$$ (which we can prove is independent of the choice of $$x$$ and local equations). Then we define the $$\textbf{intersection multiplicity}$$ of $$D_1,\ldots,D_j$$ at $$C_i$$ as the length of this set as an $$\mathcal O_{X,{C_i}}$$-module. We can also define the $$\textbf{intersection number}$$ of $$D_1,\ldots,D_j$$ as the length of the above set as a $$k$$-module (i.e., dimension as $$k$$-vector space).

My question is: are these notions the same thing? Seems like not because $$\mathcal O_{C_i}$$ is a much bigger set than $$k$$. On page 239 of the book Shafarevich seems to claim that these notions are the same at least in the case $$k=n$$, although he only makes this claim before defining intersection multiplicities. More specifically, my question is if we can prove or disprove $$\ell_k(\mathcal O_{C_i}/(f_1,\ldots,f_j))=\ell_{\mathcal O_{C_i}}(\mathcal O_{C_i}/(f_1,\ldots,f_j)),$$ at least in the case $$j=n$$, where $$C_i$$ would just be a point (where $$\ell_R(M)$$ is the length of the $$R$$-module $$M$$ for a ring $$R$$).

I managed to see that any $$\mathcal O_{C_i}$$-submodule of $$\mathcal O_{C_i}/(f_1,\ldots,f_j)$$ is a $$k$$-submodule, but I am struggling with showing the opposite.

• Where are you even seeing a definition for the intersection number of $j < n$ divisors? Commented Jun 9, 2022 at 0:47
• Go to page 238 (page 254 of the pdf). Shafarevich uses intersection numbers of $n-1$ divisors in order to prove that the intersection number of $n$ divisors is constant under linear equivalence. Commented Jun 9, 2022 at 2:46
• Ah, the intersection multiplicity of $j$ divisors along a component. The term "intersection number" is reserved for the intersection of $n$ divisors. Commented Jun 9, 2022 at 3:33
• Ah, I didn't realize that was a different name! Very confusingly written book... Seems like shafarevich uses the term interchangebly, even though it looks like we can generalize to two different notions of intersection. Commented Jun 9, 2022 at 4:40

It's not true that every $$k$$-submodule is an $$\mathcal{O}_{C_i}$$-submodule. For example, the subring $$k \subset k[x]/x^2$$ is a vector subspace but not a submodule.
Claim. Let $$k$$ be a field, $$R$$ a local noetherian $$k$$-algebra with maximal ideal $$\mathfrak{m}$$ and assume $$R/\mathfrak{m} = k$$. If $$M$$ is a finite-length $$R$$-module, then $$\ell_R(M) = \dim_k(M)$$.
Proof. If $$M=0$$, both sides are $$0$$. Otherwise, let $$\mathfrak{m}$$ be the maximal ideal of $$R$$. We have $$0 \to \mathfrak{m}M \to M \to M/\mathfrak{m}M \to 0,$$ so $$\ell_R(M) = \ell_R(\mathfrak{m}M) + \ell_R(M/\mathfrak{m}M)$$. By Nakayama's lemma, $$M/\mathfrak{m}M \ne 0$$ since $$M \ne 0$$, so $$\ell_R(M/\mathfrak{m}M) > 0$$. Also, $$M/\mathfrak{m}M$$ is now an $$R/\mathfrak{m}$$-module, so $$\ell_R(M/\mathfrak{m}M) = \ell_{R/\mathfrak{m}}(M/\mathfrak{m}M) = \ell_k(M/\mathfrak{m}M) = \dim_k(M/\mathfrak{m}M).$$ By induction, $$\ell_R(\mathfrak{m}M) = \dim_k(\mathfrak{m}M)$$, so we're done.