Lagmental Vicfred

Going Up Lifts Chains of Prime Ideals by Vicfred

tags: mathematics, integral extensions, normalization, spectra

Integral extensions allow a prime above the bottom of a chain to be extended above the top. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Definitions first

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.

$$ \mathfrak p_1\subseteq\mathfrak p_2\subseteq A,\qquad\mathfrak q_1\cap A=\mathfrak p_1 $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \exists\mathfrak q_2\supseteq\mathfrak q_1,\qquad\mathfrak q_2\cap A=\mathfrak p_2 $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \begin{array}{ccc}\mathfrak q_1&\subseteq&\mathfrak q_2\\\cap&&\cap\\\mathfrak p_1&\subseteq&\mathfrak p_2\end{array} $$

The reusable statement

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

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathfrak p_1\subseteq\mathfrak p_2\subseteq A,\qquad\mathfrak q_1\cap A=\mathfrak p_1,\\[5pt] \mathsf{C}\;&:\quad \exists\mathfrak q_2\supseteq\mathfrak q_1,\qquad\mathfrak q_2\cap A=\mathfrak p_2. \end{aligned} $$

A nearby false statement

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists\mathfrak q_2\supseteq\mathfrak q_1,\qquad\mathfrak q_2\cap A=\mathfrak p_2 \end{gathered}} $$

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 Thu 09 July 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.