On the integral degree of integral ring extensions

TL;DR

This paper investigates the integral degree invariant in integral ring extensions, proving its sub-multiplicativity and upper-semicontinuity under specific conditions, and exploring its general behavior.

Contribution

It introduces and studies the properties of the integral degree invariant, establishing conditions for sub-multiplicativity and semicontinuity, and analyzing its behavior in various cases.

Findings

The integral degree is sub-multiplicative in simple extensions.

It is upper-semicontinuous when the extension is simple, projective, finite, or the base is integrally closed.

In general, the integral degree may not be sub-multiplicative or semicontinuous.

Abstract

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\rm d}_A(B)$, is defined as the supremum of the degrees of minimal integral equations of elements of $B$ over $A$. It is an invariant that lies in between ${\rm d}_K(L)$ and $\mu_A(B)$, the minimal number of generators of the $A$-module $B$. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if $A\subset B$ is simple; if $A\subset B$ is projective and finite and $K\subset L$ is a simple algebraic field extension; or if $A$ is integrally closed. Furthermore, ${\rm d}$ is semicontinuous if $A$ is noetherian of dimension $1$ and with finite integral closure. In general, however, ${\rm d}$ is neither sub-multiplicative nor upper-semicontinuous.