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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.

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    $\begingroup$ Where are you even seeing a definition for the intersection number of $j < n$ divisors? $\endgroup$ Commented Jun 9, 2022 at 0:47
  • $\begingroup$ 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. $\endgroup$ Commented Jun 9, 2022 at 2:46
  • $\begingroup$ Ah, the intersection multiplicity of $j$ divisors along a component. The term "intersection number" is reserved for the intersection of $n$ divisors. $\endgroup$ Commented Jun 9, 2022 at 3:33
  • $\begingroup$ 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. $\endgroup$ Commented Jun 9, 2022 at 4:40

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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.

Rather, the claim is just that there are some submodules that work, enough to prove that the length equals the dimension:

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.

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