2015/07/26 by J.-F. Pommaret, Pommaret, Jean-François
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #Differential Geometry (math.DG) #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1507.07233
openalex publication_date 2015/07/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 \mathfraka ⊂ k[χ_1,..., χ_n]=k[χ] 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 \mathfrakq with associated prime ideal \mathfrakp = rad(\mathfrakq) is such that any ideal \mathfraka of the principal class with \mathfraka ⊂ \mathfrakq determines a primary ideal of greater \it multiplicity over k. In particular, we have dim_k(k[χ]/(χ_1,...,χ_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 \mathfraka of the principal class with \mathfraka⊂ (χ_1,â,χ_n)2 is contained into a \it simple ideal, that is a primary ideal \mathfrakq such that rad(\mathfrakq)=\mathfrakm∈ max(k[χ]) is a maximal and thus prime ideal with dim_k(M)=dim_k(k[χ]/\mathfrakq)=2n at least. Accordingly, any primary ideal \mathfrakq may not be a member of the primary decomposition of an unmixed ideal \mathfraka ⊆ \mathfrakq of the principal class. Otherwise, \mathfrakq 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.