Lagmental Vicfred

Intersection Multiplicity Is a Local Algebra Length by Vicfred

tags: mathematics, smoothness, singularities, etale morphisms

At an isolated intersection, the quotient by both local equations measures tangency and multiplicity. A small computation will anchor the general statement before the abstraction takes over.

Notation

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.

$$ I_p(C,D)=\dim_k\mathcal O_{\mathbf A^2,p}/(f,g) $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ I_p(C,D)=1\Longleftrightarrow C,D\ \text{meet transversely at }p $$

Stress the formula

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ f=y,\quad g=y-x^m,\quad p=(0,0)\Longrightarrow\frac{k[x,y]_{(x,y)}}{(y,y-x^m)}\cong\frac{k[x]_{(x)}}{(x^m)},\quad I_p=m $$

Interpretation

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad I_p(C,D)=\dim_k\mathcal O_{\mathbf A^2,p}/(f,g),\\[5pt] \mathsf{C}\;&:\quad I_p(C,D)=1\Longleftrightarrow C,D\ \text{meet transversely at }p. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] I_p(C,D)=1\Longleftrightarrow C,D\ \text{meet transversely at }p \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 26 August 2013. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.