Lagmental Vicfred

The Zariski Tangent Space Is a Kernel of the Jacobian by Vicfred

tags: mathematics, smoothness, singularities, etale morphisms

At a rational point, tangent vectors solve the linearized defining equations. This is a compact note, but the quantifiers and hypotheses stay on the page.

Statement

For \(X=V(f_1,\ldots,f_r)\subseteq\mathbf A^n\), the Jacobian matrix \(J_p=(\partial f_i/\partial x_j)(p)\) controls tangent dimensions. Smoothness asks for the expected rank after passing to the residue field.

$$ J_p=\left(\frac{\partial f_i}{\partial x_j}(p)\right)_{i,j} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ T_pX=\ker(J_p:k^n\to k^r) $$

Worked algebra

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ f=y^2-x^3,\quad J_{(0,0)}=\begin{pmatrix}0&0\end{pmatrix},\qquad T_{(0,0)}X=k^2 $$

Conceptual compression

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad J_p=\left(\frac{\partial f_i}{\partial x_j}(p)\right)_{i,j},\\[5pt] \mathsf{C}\;&:\quad T_pX=\ker(J_p:k^n\to k^r). \end{aligned} $$

Caveat

A visually sharp point need not capture scheme-theoretic singularity, and characteristic can make every partial derivative vanish unexpectedly.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] T_pX=\ker(J_p:k^n\to k^r) \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Wed 05 October 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.