Lagmental Vicfred

Lying Over Makes an Integral Map of Spectra Surjective by Vicfred

Last updated: Sun 17 July 2016

tags: mathematics, integral extensions, normalization, spectra

Every prime of the base ring has a prime above it in an integral extension. The point is to make the formal expression readable enough to audit line by line.

The data

An element \(b\) is integral over \(A\) when it satisfies a monic polynomial with coefficients in \(A\). An extension \(A\subseteq B\) is integral when every \(b\in B\) has this property.

$$ A\subseteq B\ \text{integral},\qquad\mathfrak p\in\operatorname{Spec}A $$

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.

$$ \exists\mathfrak q\in\operatorname{Spec}B,\qquad\mathfrak q\cap A=\mathfrak p $$

Derivation

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

$$ \operatorname{Spec}B\xrightarrow{\ \pi\ }\operatorname{Spec}A,\qquad\pi(\mathfrak q)=\mathfrak q\cap A,\qquad\pi\ \text{surjective} $$

Invariant content

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.

$$ \begin{aligned} \mathsf{D}\;&:\quad A\subseteq B\ \text{integral},\qquad\mathfrak p\in\operatorname{Spec}A,\\[5pt] \mathsf{C}\;&:\quad \exists\mathfrak q\in\operatorname{Spec}B,\qquad\mathfrak q\cap A=\mathfrak p. \end{aligned} $$

Scope

Algebraic over a fraction field and integral over the base ring are different conditions. Denominators make many algebraic elements nonintegral.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists\mathfrak q\in\operatorname{Spec}B,\qquad\mathfrak q\cap A=\mathfrak p \end{gathered}} $$

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 Fri 06 March 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.