Lagmental Vicfred

Étale Morphisms Are Algebraic Local Isomorphisms by Vicfred

tags: mathematics, smoothness, singularities, etale morphisms

A morphism is étale when it is flat, unramified, and locally of finite presentation. 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.

$$ f:X\to Y\ \text{étale} $$

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.

$$ \Omega_{X/Y}=0,\qquad f\ \text{flat and locally of finite presentation} $$

Stress the formula

The middle display is intentionally dense: it is where signs, bounds, multiplicities, or normalising factors are most likely to be lost.

$$ B=A[t]/(g),\qquad g'(\bar t)\in B^\times\Longrightarrow\operatorname{Spec}B\to\operatorname{Spec}A\ \text{étale} $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f:X\to Y\ \text{étale},\\[5pt] \mathsf{C}\;&:\quad \Omega_{X/Y}=0,\qquad f\ \text{flat and locally of finite presentation}. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \Omega_{X/Y}=0,\qquad f\ \text{flat and locally of finite presentation} \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sun 29 March 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.