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Exercise 5.4 of Hartshorne defines the intersection multiplicity $(Y \cdot Z)_p$ of two distinct plane curves $Y = Z(f), Z = Z(g) \subset \mathbb A^2$ in $p \in Y \cap Z$ as the length of the $\mathcal O_p$-module $\mathcal O_p/(f,g)$.

I'm trying to understand it with a simple example: $$f = x^2 - y^2 + x^4, \qquad g = y$$ I think that $$k[x,y]/(x^2-y^2+x^4,y) \simeq k[x]/(x^2+x^4)$$ and that as $k$-algebras $$\mathcal O_p/(f,g) \simeq k[x]_{(x)}/(x^2+x^4) \simeq \big( k[x]/(x^2+x^4) \big)_{(x)}$$ but I don't know if that's really supposed to help me or not. How can I proceed? Is there a better approach?

woolly-minded
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Your reasoning is right, you just have to conclude noting that, localizing at $(x)$ makes $1+x^2$ a unit, and so $$ \big( k[x]/(x^2+x^4) \big)_{(x)} = \big( k[x]/(x^2(1+x^2)) \big)_{(x)} \cong \big( k[x]/(x^2) \big)_{(x)} \cong k[x]/(x^2) $$ has length 2.

KReiser
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Aitor Iribar Lopez
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  • In what sense does it have length 2? Still as an O_p-module? – woolly-minded May 08 '22 at 18:51
  • If $A$ is a ring and $I$ and ideal of $A$, the length of $A/I$ as an $A$-module is the same as its length as an $A/I$-module (because the $A$-submodules and the $A/I$-submodules are the same) – Aitor Iribar Lopez May 08 '22 at 20:09
  • Then, can you tell me if the following is correct? We're looking for the Op-length of Op/(f,g), but that's the same thing as its Op/(f,g)-length. But that means we're just studying its ideals, and as https://math.stackexchange.com/questions/375353/every-ideal-of-the-localization-is-an-extended-ideal#375359 shows, we can study chains of ideals of k[x,y]/(f,g) and then see how those chains behave in the localization. – woolly-minded May 09 '22 at 18:24
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    Yes, it is correct. In fact the ring $k[x,y]/(f,g)$ is Artinian if $f$ and $g$ are coprime, and so it is a direct sum of artinian local rings $A_1 \oplus \ldots \oplus A_n$. Each of these rings is isomorphic to $\mathcal O_p/(f,g)$ for the common zeros of $f$ and $g$ so their length is the multiplicity. In the case you wrote, $k[x,y]/(f,g) \cong k[x]/(x^4+x^2) \cong k[x]/(x^2) \oplus k[x]/(x+i) \oplus k[x]/(x-i)$ where $i \in k$ is such that $i^2 = -1$. – Aitor Iribar Lopez May 09 '22 at 21:45