Pure Differential Modules and a Result of Macaulay on Unmixed Polynomial Ideals

TL;DR

This paper revisits Macaulay's 1916 result on the Hilbert function of differential modules, clarifies its modern interpretation, and provides examples illustrating properties of primary ideals and symbols related to acyclicity and involutivity.

Contribution

It offers a modern explanation of Macaulay's classical result and constructs new examples demonstrating properties of differential modules and ideals.

Findings

Exhibits symbols that are 2, 3, 4-acyclic but not involutive

Provides examples similar to conformal Killing systems

Clarifies the structure of primary ideals in differential modules

Abstract

The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \subset k[{\chi}\_1$,..., ${\chi}\_n]=k[\chi]$ is of the {\it principal class} and thus {\it unmixed} if it has rank $r$ and is generated by $r$ polynomials. Having in mind this definition, a primary ideal $\mathfrak{q}$ with associated prime ideal $\mathfrak{p} = rad(\mathfrak{q})$ is such that any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a} \subset \mathfrak{q}$ determines a primary ideal of greater {\it multiplicity} over $k$. In particular, we have $dim\_k(k[\chi]/({\chi}\_1$,...,${\chi}\_n)^2)=n+1$ because, passing to a system of PD equations for one unknown $y$, the parametric jets are \{${y,y\_1, ...,y\_n}$\} but any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a}\subset ({\chi}\_1,{\^a},{\chi}\_n)^2$ is contained into a {\it simple} ideal, that is a primary ideal $\mathfrak{q}$ such that $rad(\mathfrak{q})=\mathfrak{m}\in max(k[\chi])$ is a maximal and thus prime ideal with $dim\_k(M)=dim\_k(k[\chi]/\mathfrak{q})=2^n$ at least. Accordingly, any primary ideal $\mathfrak{q}$ may not be a member of the primary decomposition of an unmixed ideal $\mathfrak{a} \subseteq \mathfrak{q}$ of the principal class. Otherwise, $\mathfrak{q}$ is said to be of the {\it principal noetherian class} ". Our aim is to explain this result in a modern language and to illustrate it by providing a similar example for $n=4$. The importance of such an example is that it allows for the first time to exhibit symbols which are $2,3,4$-acyclic without being involutive. Another interest of this example is that it has properties quite similar to the ones held by the system of conformal Killing equations which are still not known. For this reason, we have put all the examples at the end of the paper and each one is presented in a rather independent way though a few among them are quite tricky. Meanwhile, the second purpose is to prove that the methods developped by Macaulay in order to study {\it unmixed polynomial ideals} are only particular examples of new formal differential geometric techniques that have been introduced recently in order to study {\it pure differential modules}. However these procedures are based on the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations, in particular on a systematic use of the Spencer operator, and are still not acknowledged by the algebraic community.