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I want to do a concrete example of an intersection product for myself.

Consider the endomorphism $f:\mathbf{P}^1_k\to \mathbf{P}^1_k$ given by $(x:y)\to (y:x)$. It has precisely two fixed points: $(1:1)$ and $(1:-1)$. I want to compute that the intersection product on $\mathbf{P}^1\times \mathbf{P}^1$ is two.

I think I can somehow see that this is the length of the module $k[x,y]/(x-y,x+y)$...Can somebody explain or correct me?

I also wish I was capable of doing something similar with the morphism $(x:y)\mapsto (x^n:y^n)$. Can somebody explain how this works?

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