Lagmental Vicfred

Affine Morphisms Reverse Ring Maps by Vicfred

tags: mathematics, schemes, morphisms, functor of points

A morphism between affine schemes corresponds contravariantly to one homomorphism of coordinate rings. A small computation will anchor the general statement before the abstraction takes over.

Start locally

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.

$$ \varphi:A\to B $$

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 \(\Longleftrightarrow\).

$$ \operatorname{Spec}B\xrightarrow{\ \varphi^\ast\ }\operatorname{Spec}A,\qquad\mathfrak q\mapsto\varphi^{-1}(\mathfrak q) $$

Compute before generalising

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \begin{array}{ccc}A&\xrightarrow{\varphi}&B\\\operatorname{Spec}A&\xleftarrow{\varphi^\ast}&\operatorname{Spec}B\end{array} $$

The global view

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad \varphi:A\to B,\\[5pt] \mathsf{C}\;&:\quad \operatorname{Spec}B\xrightarrow{\ \varphi^\ast\ }\operatorname{Spec}A,\qquad\mathfrak q\mapsto\varphi^{-1}(\mathfrak q). \end{aligned} $$

Edge conditions

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \operatorname{Spec}B\xrightarrow{\ \varphi^\ast\ }\operatorname{Spec}A,\qquad\mathfrak q\mapsto\varphi^{-1}(\mathfrak q) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Sun 16 August 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.