Noether normalizations, reductions of ideals, and matroids

TL;DR

This paper introduces a unifying framework called generic matroids to understand the exchange properties of Noether normalizations and reductions of ideals in algebraic structures, generalizing classical concepts.

Contribution

It establishes generic exchange theorems for Noether normalizations and reductions, and introduces generic matroids as a new topological-combinatorial structure.

Findings

Proves exchange properties for Noether normalizations in graded algebras.

Establishes analogous exchange theorems for ideal reductions.

Introduces the concept of generic matroids as a unifying framework.

Abstract

We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which is a common generalization of a topological space and a matroid.