Let $V$ be a non-singular affine variety in $\mathbb{C}^n$.
$V$ can be regarded as a complex manifold.
Let $p = (a_1,\dots,a_n) $ be a point of $V$.
Let $\mathcal{O}_p$ be the local ring of $V$ at $p$.
A tangent vector $v$ at $p$ is a derivation $\mathcal{O}_p \rightarrow \mathbb{C}$, i.e.
a $\mathbb{C}$-linear map $v$ such that $v(fg) = v(f)g(p) + f(p)v(g)$ for $f, g \in \mathcal{O}_p$.
Let $T_p$ be the set of tangent vectors at $p$.
We regard $T_p$ as a vector space over $\mathbb{C}$ in the obvious way.
On the other hand, we can define a tangent space at $p$ as follows.
Let $f_1,\dots f_r$ be defining polynomials for $V$.
Let $L_i$ be the hyperplane defined by $\sum_k \frac{\partial f_i}{\partial x_k}(p)(x_k – a_k) = 0$.
Let $S_p = \bigcap_i L_i$.
Are $T_p$ and $S_p$(or rather the vector space attached to it) canonically isomorphic?
Best Answer
Yes, there is a canonical isomorphism of $\mathbb C$-vector spaces $$i:S_{X,p}\stackrel {\cong}{\to} T_{X,p}=Der(\mathcal O_{X,p}, \mathbb C)$$ If one starts with a vector $v=(v_1,...,v_n)\in S_p\subset \mathbb C^n$ (so that $\sum_k \frac{\partial f_i}{\partial x_k}(p)v_k = 0$), the isomorphism $i$ associates to it the derivation $i(v)=\partial _v$ defined on $\mathcal O_{X,p}$ by the formula $$ \partial_v(g)=\sum_k \frac{\partial g}{\partial x_k}(p)v_k $$ where $g\in \mathcal O_{X,p}$ is an arbitrary local function.
The inverse isomorphism is given by $$i^{-1}: Der(\mathcal O_{X,p},\mathbb C) \stackrel {\cong}{\to} S_{X,p} :\partial \mapsto (\partial x_1,...,\partial x_n) $$
Edit
The vector space of derivations is also isomorphic to Zariski's tangent space $(\frak m_p/\frak m^2_p)^*$, defined via the maximal ideal $\mathfrak m_p\subset \mathcal O_{X,p}$.
The isomorphism is $$ Der(\mathcal O_{X,p} ) \stackrel {\cong}{\to} (\frak m_p/\frak m^2_p)^*:\partial \mapsto \overline{\partial} $$ where $\overline{\partial} (g \;\text {mod} \;m^2_p)=\partial (g)$ for $g\in \frak m_p$.
Second Edit
The following remark may be of some interest, since it does not seem to be addressed in Algebraic Geometry books:
If $X\subset \mathbb A^n_k$ is an affine algebraic variey and if $p\in X$, we may consider the maximal ideal $M_p\subset \mathcal O(X)$ of global functions vanishing at $p$.
We may also consider, as we already did, the maximal ideal $\mathfrak m_p\subset \mathcal O_{X,p}$ of germs of functions regular ay $p$ and vanishing at $p$.
We then have a natural $k$-linear map $ M_p/M_p^2 \to \mathfrak m_p/\mathfrak m_p^2 $ and the slightly surprising but pleasant fact is that this linear map is an isomorphism.