# the differential of a regular map between varieties

I'm reading the book of Shafarevich of algebraic geometry. I'm having troubles with this page: What is explicitly the differential of a map in terms of the regular map $f:X\to Y$? Obviously is a composition of a lot of maps , isomorphism between the tangent space and $m_x/m^2_x$, etc... I think that it's very obvious but I prefer to be sure.

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## 3 Answers

Typing "Differential of a regular map" into Google gave me this:

http://www.jmilne.org/math/CourseNotes/AG500.pdf (I recommend pages 78+)

Typing "Differential of a smooth map" into Google gave me this:

https://en.wikipedia.org/wiki/Pushforward_(differential)

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Yes... It's similar but it's not the same, here we are working over any kind of algebraically closed field, and not necessary over the real numbers. We are working over Varieties, in Algebraic Geometry and not manifolds. –  Daniel Jan 28 '13 at 21:40

Technically, given a linear form $\ell\in(\mathfrak m_x/\mathfrak m_x^2)^\ast$, we have

$$(d_xf)(\ell)=\Bigl(\mathfrak m_y/\mathfrak m_y^2\ni \phi \mapsto \ell(f^\ast(\phi))\Bigr)$$

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Given $f: X \to Y$ we go $$\Theta_{X, x} = \text{Hom}(\mathfrak{m}_x/\mathfrak{m}_x^2, k) \stackrel{?}{\to}\text{Hom}(\mathfrak{m}_y/\mathfrak{m}_y^2, k) = \Theta_{Y, y}.$$

Here the maps which I'm calling = are the content of Corollary 1, and the map I'm calling $?$ is $\phi \mapsto \phi \circ f^*$ where $f^*$ is the usual pullback map on local rings $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$.

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