# Motivation behind the definition of Zariski tangent space

Intuitively I think of tangent space at a point as the set of all points lying in the tangent plane passing throug that point.

Here is the definition of Zariski tangent space

Let X be an algebraic variety. and $p \in X$. The tangent space of $X$ at point $p$ is defined as $$T_pX= Der_k(O_{X,p}.k)$$

How does the above definition match with my intuition? Or more specifically, can someone give a one-one correspondence between $T_pX$ and the set of all points lying in the tangent plane passing through $p$?

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This definition (or something very close) is fairly standard in differential geometry. It might be worth opening up a differential geometry book and seeing why this definition is equivalent to more intuitive definitions of the tangent space of a smooth manifold. –  Brett Frankel Feb 11 '13 at 15:55
You can also write down a formal definition for your "intuitive" tangent space and then prove that the vector spaces you get are isomorphic. It's not too hard, though there is a trick involved. –  Zhen Lin Feb 11 '13 at 16:00
What will be the formal definition of my intuitive tangent space? –  Mohan Feb 11 '13 at 16:11
I believe it's in Shafarevich, but I don't know for sure. Just take $X$ to be an affine variety embedded in affine space and write down the obvious thing associated with your mental picture. –  Zhen Lin Feb 11 '13 at 18:31

The tangent space at $p$ is the space of all directions in which you can take a directional derivative at $p$. Whatever "directional derivative" means, it should only depend on the germ at $p$, so it's a function on $\mathcal{O}_{X, p}$. And it should be linear and obey the Leibniz rule, so it's a derivation. These conditions turn out to be enough to give a notion of tangent space that agrees with intuition (e.g. you can compute the Zariski tangent space to a variety cut out by various polynomials and it will be the thing you think it is).

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Second remark: you didn't specify whether the field has characteristic zero or not...I assume it does. Third remark: what you wrote down isn't actually the definition of the Zariski tangent space. The definition of the Zariski tangent space is $m_p/m_p^2$ where $m_p$ is the ideal of all algebraic functions which vanish at p.