An Algebraic Generalization of Subelliptic Multipliers

Abstract

In this thesis, we are primarily concerned with an algorithm, due to Kohn, for finding subelliptic multipliers in the theory of several complex variables. Kohn's work applies to rings of germs of smooth functions. A special case of it leads to an interesting algebraic procedure defined for the ring of convergent power series with complex coefficients. We modify this procedure by extending it to more general regular local rings. To do so, we define two nonlinear operations between the set of modules of 1-forms and the set of ideals, and we alternatively apply these operations to a given initial module of 1-forms to obtain an increasing sequence of modules of 1-forms. In case that a regular local ring satisfies the weak Jacobian condition at the maximal ideal over a quasi-coefficient field of characteristic zero, we construct subelliptic multipliers for a submodule of the universally finite module of differentials on the given ring. In particular we provide an unusual process for determining whether an ideal is primary to the maximal ideal.