Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm reading the book of Shafarevich of algebraic geometry. I'm having troubles with this page: enter image description here 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.

share|cite|improve this question

Typing "Differential of a regular map" into Google gave me this: (I recommend pages 78+)

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

share|cite|improve this answer
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

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

share|cite|improve this answer

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 $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.