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I'm struggling with the following exercise I need to solve and present in our Algebraic Geometry class (book: Algebraic Curves, William Fulton, 2008 Edition, Exercise 2.22):

Let $k$ be an algebraically closed field, $\mathbb{A}^n$ the cartesian product of $k$ with itself $n$ times. $\mathcal{O}_p(V)$ is the set of rational functions on a subvariety $V$ of $\mathbb{A}^n$ that are defined at the point $p \in V$.

Further, let $T: \mathbb{A}^n \to \mathbb{A}^n$ be an affine change of coordinates with $T(p) = q$.

I need to show two things:

  1. $\tilde{T}: \mathcal{O}_q(\mathbb{A}^n) \to \mathcal{O}_p(\mathbb{A}^n)$ is an isomorphism.

  2. $\tilde{T}$ induces an isomorphism from $\mathcal{O}_q(V) \to \mathcal{O}_p(T^{-1}(V))$ if $p \in T^{-1}(V)$, for $V$ a subvariety of $\mathbb{A}^n$.

I really don't know how to begin. Do I have to find an explicit map and show it is an isomorphism or is there a trick?

Thank you very much for your help in advance!

jk

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    $\begingroup$ Isn't $\tilde{T}: \mathcal{O}_q(\mathbb{A}^n) \to \mathcal{O}_p(\mathbb{A}^n)$ just given by $f \mapsto f \circ T$? Doesn't precomposition by $T^{-1}$, i.e. $g \mapsto g \circ T^{-1}$ give the inverse homomorphism $\tilde{T}^{-1}: \mathcal{O}_p(\mathbb{A}^n) \to \mathcal{O}_q(\mathbb{A}^n)$? $\endgroup$ – André 3000 Mar 27 '15 at 3:09

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