Lagmental Vicfred

A Scheme Point Carries a Residue Field by Vicfred

tags: mathematics, schemes, morphisms, functor of points

The residue field at a prime records the smallest field through which that point factors. A small computation will anchor the general statement before the abstraction takes over.

Set-up

An affine scheme \(X=\operatorname{Spec}A\) is determined by its ring together with its prime spectrum and structure sheaf. Maps \(X\to Y\) reverse the direction of ring homomorphisms.

$$ x\leftrightarrow\mathfrak p\in\operatorname{Spec}A $$

There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.

$$ \kappa(x)=\operatorname{Frac}(A/\mathfrak p)=A_{\mathfrak p}/\mathfrak pA_{\mathfrak p} $$

The calculation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ A=k[t],\qquad\begin{cases}\mathfrak p=(t-a)&\Longrightarrow\kappa(\mathfrak p)=k,\\\mathfrak p=(0)&\Longrightarrow\kappa(\mathfrak p)=k(t).\end{cases} $$

What survives abstraction

What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.

$$ \begin{aligned} \mathsf{D}\;&:\quad x\leftrightarrow\mathfrak p\in\operatorname{Spec}A,\\[5pt] \mathsf{C}\;&:\quad \kappa(x)=\operatorname{Frac}(A/\mathfrak p)=A_{\mathfrak p}/\mathfrak pA_{\mathfrak p}. \end{aligned} $$

The boundary

Field-valued points see only part of a scheme. Nilpotents, residue-field extensions, and families over nonfields require general test rings.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \kappa(x)=\operatorname{Frac}(A/\mathfrak p)=A_{\mathfrak p}/\mathfrak pA_{\mathfrak p} \end{gathered}} $$

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 Sun 09 August 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.