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Let $X,Y$ be curves of $\mathbb{A}^2$ given by irreducible polynomials $f,g$ respectively, where the ground field $k$ is algebraically closed. Then it is known that the dimension of $k[x,y]/(f,g)$ as a $k$-vector space is equal to the cardinality of $X \cap Y$.

Now let $P \in X \cap Y$. The intersection multiplicity of $X,Y$ at $P$ is defined as the dimension of the $k$-vector space $\mathcal{O}_P/(f,g)$, where $\mathcal{O}_P$ is the local ring at $P$. Intuitively, the effect of localization is that of removing contributions of points in $X \cap Y$ other than $P$. Question 1: How can we see that rigorously?

In particular, it is mentioned here (top of page 3) http://www.math.lsa.umich.edu/~hochster/cmrvw.ps that "it suffices to invert enough elements of $k[x,y]$ such that for any point in $V(f,g)-P$, at least one of the inverted elements vanishes at that point. Question 2: Why is it enough to do that? I.e. why inverting such elements removes the effect of other points in $X \cap Y$?

Question 3: How can we see that $\dim_k \mathcal{O}_P/(f,g) \ge \mu_P(X) \mu_P(Y)$? This is problem I.5.4.(a) in Hartshorne and i have found none of the available online solutions satisfactory.

Edit:

Definition: $\mu_P(X)$ is the multiplicity of point $P$ in the variety $X$. It can be computed as follows: do a change of coordinates such that $P$ maps to $(0,0)$ and write $f=f_r+f_{r+1}+\cdots+f_n$, where $f_i$ is homogeneous of degree $i$. Then $\mu_P(X)=r$.

Manos
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  • For your first two questions, this is purely intuition. Is this (http://math.stackexchange.com/questions/882382/localization-of-the-integer-ring/882390#882390) at all helpful? If not, I can try and explain more. For the third question, would you mind defininig $\mu_p$? – Alex Youcis Aug 23 '14 at 03:17
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    On Q1: $\mathscr{O}_P / (f, g)$ is isomorphic to a localisation of $k [x, y] / (f, g)$: more precisely, it is isomorphic to the ring obtained by inverting in $k [x, y] / (f, g)$ the image of any function not vanishing at $P$. But $k [x, y] / (f, g)$ is (usually) artinian, hence is isomorphic to the direct product of some number of copies of $k [\epsilon] / (\epsilon^n)$; what this localisation does is to extract the copy corresponding to $P$. – Zhen Lin Aug 24 '14 at 19:08
  • @ZhenLin: How do you see that $k[x,y]/(f,g)$ has this structure (if it is Artinian)? – Manos Aug 25 '14 at 13:31
  • @ZhenLin: I can see that the ring is a direct product of local rings, each corresponding to a point of the intersection, however why each factor in the product has the form $k[\epsilon]/(\epsilon^n)$? – Manos Aug 25 '14 at 14:04
  • @AlexYoucis: I included a definition of $\mu_P$. Zhen Lin's comment is along the lines of the answer i was looking for. If you can add any further intuition on that it would be great. – Manos Aug 25 '14 at 14:12
  • The number of generators is related to the dimension of the intersection of the tangent spaces. Since we are dealing with curves, that means the number of generators is at most one. – Zhen Lin Aug 25 '14 at 14:23
  • @ZhenLin: Could you describe this relation more explicitely? – Manos Aug 25 '14 at 14:32
  • @ZhenLin: Or provide a reference... – Manos Aug 25 '14 at 14:49

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