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For $X$ an affine variety and $p \in X$ define $T_p X = \mathrm{Der}(k[X], \mathrm{ev}_p)$.

Claim: If $Y = X \setminus Z(f)$ is some Zariski open affine subvariety of $X$ and $p \in Y$, then $T_p X \cong T_pY$.

Definition: Let $X$ be an arbitrary variety, $p \in X$. Define $T_p X = T_p U$ for any open affine neighbourhood $U$ of $p$.

Apparently the above claim tells us this definition makes sense, but I can't see why. I'd appreciate an explanation (note that I'm only really concerned with projective varieties, so if a simpler explanation can be afforded by restricting to this case then please do so!)

Thanks

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    http://math.stackexchange.com/questions/144202/tangent-space-at-some-point-of-a-quasi-projective-variety2012-06-02
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    Let $U$ and $V$ be two open affine neighborhoods of $p$ ... and therefore $T_pU = T_pV$.2012-06-02
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    @Hurkyl I must be missing something. Why?2012-06-02
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    @Matt: The intersection of two open affine neighbourhoods contains an open affine neighbourhood.2012-06-02
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    @ZhenLin I don't see why we can restrict to the case $ Y = X \setminus Z(f)$...2012-06-02
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    Every open affine neighbourhood contains an open affine neighbourhood _of that form_.2012-06-02
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    Distinguished open subsets are a base for the Zariski topology.2012-06-02

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