Lagmental Vicfred

Closed Immersions Are Surjections of Coordinate Rings by Vicfred

Last updated: Fri 23 September 2022

tags: mathematics, schemes, morphisms, functor of points

An affine closed subscheme is obtained by quotienting the ambient coordinate ring by an ideal. The point is to make the formal expression readable enough to audit line by line.

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.

$$ i:Z\hookrightarrow X=\operatorname{Spec}A $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ Z\cong\operatorname{Spec}(A/I),\qquad A\twoheadrightarrow A/I $$

Stress the formula

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

$$ \left.\begin{aligned}I&=(y^2-x^3),\\Z&=\operatorname{Spec}k[x,y]/I\end{aligned}\right\}\hookrightarrow\mathbf A_k^2 $$

Interpretation

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad i:Z\hookrightarrow X=\operatorname{Spec}A,\\[5pt] \mathsf{C}\;&:\quad Z\cong\operatorname{Spec}(A/I),\qquad A\twoheadrightarrow A/I. \end{aligned} $$

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.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] Z\cong\operatorname{Spec}(A/I),\qquad A\twoheadrightarrow A/I \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Mon 26 November 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.