Deformations of the Exterior Algebra of Differential Forms

Abstract

Let $D:\Omega\xrightarrow{}\Omega$ be a differential operator defined in the exterior algebra $\Omega$ of differential forms over the polynomial ring $S$ in $n$ variables. In this work we give conditions for deforming the module structure of $\Omega$ over $S$ induced by the differential operator $D$, in order to make $D$ an $S$-linear morphism while leaving the $\mathbb{C}$-vector space structure of $\Omega$ unchanged. One can then apply the usual algebraic tools to study differential operators: finding generators of the kernel and image, computing a Hilbert polynomial of these modules, etc. Taking differential operators arising from a distinguished family of derivations, we are able to classify which of them allow such deformations on $\Omega$. Finally we give examples of differential operators and the deformations that they induce.