Integral extensions and the a-invariant

TL;DR

This paper investigates the relationship between the a-invariants of a homogeneous algebra and its subalgebra, establishing conditions under which properties like minimal multiplicity are preserved in algebraic extensions.

Contribution

It provides algebraic generalizations of Hurwitz type theorems by comparing a-invariants in finite homogeneous algebra extensions, especially when one algebra is normal and the other has minimal multiplicity.

Findings

If A ⊂ B is a finite homogeneous extension with B of minimal multiplicity and A normal, then A also has minimal multiplicity.

The results generalize classical Hurwitz theorems to algebraic settings involving a-invariants.

The paper establishes conditions linking the a-invariants of subalgebras and their extensions.

Abstract

In this note we compare the a-invariant of a homogeneous algebra B to the a-invariant of a subalgebra A. In particular we show that if $A \subset B$ is a finite homogeneous inclusion of standard graded domains over an algebraically closed field with A normal and B of minimal multiplicity then A has minimal multiplicity. In some sense these results are algebraic generalizations of Hurwitz type theorems.