Lagmental Vicfred

The Functor of Points Tests a Scheme on Every Ring by Vicfred

tags: mathematics, schemes, morphisms, functor of points

A scheme can be studied through the sets of its T-valued points and their functorial dependence on T. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.

Notation

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.

$$ h_X(T)=\operatorname{Hom}(T,X) $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ X\cong Y\Longleftrightarrow h_X\cong h_Y\ \text{naturally} $$

Stress the formula

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ h_{\mathbf A^n}(T)=\operatorname{Hom}(\operatorname{Spec}T,\mathbf A^n)\cong T^n $$

Interpretation

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

$$ \begin{aligned} \mathsf{D}\;&:\quad h_X(T)=\operatorname{Hom}(T,X),\\[5pt] \mathsf{C}\;&:\quad X\cong Y\Longleftrightarrow h_X\cong h_Y\ \text{naturally}. \end{aligned} $$

Limit of the argument

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] X\cong Y\Longleftrightarrow h_X\cong h_Y\ \text{naturally} \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sat 04 January 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.