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[Springer, Linear algebraic group, 4.1.9.(3)] Let $X$ be an affine algebraic variety. Let $k[\tau]=k[T] /\left(T^{2}\right)$ be the $k$-algebra of dual numbers ($\tau$ being the image of $T$). Show there is bijection of $T_{x} X$ onto the set of $k$-homomorphisms $\phi: k[X] \rightarrow k[\tau]$ such that $\phi(f)-f(x) \in k \tau$. (These are the '$k[\tau]$-valued points of $X$ lying over $x$'.)

$T_x X \cong (M_x/M_x^2)^*$, where $M_x\subset k[X]$ is the maximal ideal of functions vanishing in $x$.

Could you please help me with this problem? Thank you in advance!

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Given $\lambda \in (M_x/M_x^2)^*$, define $\phi(f) = f(x) + \lambda(f - f(x))\tau$ for all $f \in k[X]$.

Given $\phi \mathrel: k[X] \to k[\tau]$ satisfying your condition, define $\lambda(f) = \frac1\tau\phi(f)$ for all $f \in M_x$.

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  • $\begingroup$ Thank you for your help! $\endgroup$
    – Ryze
    Jan 16, 2021 at 15:34

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