# Tangent spaces and regular maps

Let $$k$$ be an algebraically closed field. Given a morphism between $$k$$-varieties $$\phi: X \to Y$$, I want to define the induced map on the tangent spaces as follows. Consider the pullback $$\phi^\ast: \mathcal{O}_Y(U) \to \mathcal{O}_X(\phi^{-1}U)$$ for any $$U \subset Y$$ open. This yields a local homomorphism between the stalks $$\mathcal{O}_{Y, \phi(p)}$$ and $$\mathcal{O}_{X, p}$$ by sending $$[U,h]_{\phi(p)} \mapsto [\phi^{-1}U, h \circ \phi]_p,$$ where $$[U,f]_q$$ denotes a germ at $$q$$. Now, restricting to a maximal ideal $$\mathfrak{m}_{\phi(p),Y}$$, we should get a well-defined map $$\Phi: \mathfrak{m}_{\phi(p), Y}/\mathfrak{m}_{\phi(p), Y}^2 \to \mathfrak{m}_{p, X}/\mathfrak{m}_{p,X}^2.$$ and define the differential of $$\phi$$ at $$p \in X$$ as the dual map of $$\Phi$$, that is $$d_p \phi: T_pX \to T_pY, \quad \ell \mapsto \ell \circ \Phi,$$ where we use the definition of the tangent spaces as the dual space of the maximal ideals above quotient by their squares.

Question. With this definition, I would like to compute the differential of a polynomial map $$\phi: X \to Y$$ where $$X \subset \mathbb{A}^n$$ and $$Y \subset \mathbb{A}^m$$ are algebraic sets and $$\phi(x) := (f_1(x), ..., f_m(x)),$$ for polynomial functions $$f_i: k^n \to$$k. I expect it to be the multiplication by the Jacobian of $$f$$, but I have trouble to make this argument precise. So, here is my attempt (with all its mistakes): Consider the pullback $$k[y_1, ..., y_m] \to k[x_1, ...,x_n], \: y_i \mapsto y_i \circ \phi$$ which singles out the $$i$$th component of the function $$\phi$$. Since on affine varieties, the stalk is isomorphic to the localization at the respective maximal ideal, we get a map from $$\Phi_p: k[y_1, ..., y_m]_{\mathfrak{m}_{\phi(p)}} \to k[x_1, ...,x_n]_{\mathfrak{m}_p}, \quad \frac{g}{h} \mapsto \frac{g \circ \phi}{h \circ \phi}.$$ Since $$h$$ vanishes on $$\phi(p)$$, it is plain that $$h \circ \phi$$ vanishes on $$p$$, so this is well-defined. Now, take a polymomial function $$h$$ vanishing at $$p$$, that is $$h \in \mathfrak{m}_{\phi(p), Y}$$ and consider its linearization $$h^{(1)}:= \sum_{i=1}^m \frac{\partial h}{\partial y_i}\big\vert_{\phi(p)} d_{\phi(p)} y_i \in \mathfrak{m}_{\phi(p), Y}/\mathfrak{m}_{\phi(p), Y}^2.$$ So, $$\Phi_p \circ h^{(1)}= \sum_{i=1}^m \frac{\partial (h \circ \phi)}{\partial y_i}\big \vert_{\phi(p)} d_{\phi(p)}y_i = \sum_{i=1}^m \frac{\partial h}{\partial y_i}\big \vert_{\phi(p)} \frac{\partial \phi}{\partial x_i}\big \vert_{p} d_{\phi(p)}y_i = \operatorname{Jac}_f(p) \circ h^{(1)}.$$ using the chain rule on formal deriatives. So, $$\Phi_p = \operatorname{Jac}_f(p)$$. Is this correct?