Universal Modules of Differantial Operators

Abstract

This thesis is concerned with universal differential operator modules of order n. Let R be a commutative k-algebra where k is an algebraically closed field of characteristic zero. Let m and n be positive integers such that m is less than n. Then we find the generators of the kernel of the map from the universal module of nth order derivations into the universal module of mth order derivations. Then we focus on the behavior of the Betti series of the universal module of derivations. Firstly, we showed that under some conditions the Betti series of the universal module of second order derivations of the localization of R at m is rational where m is a maximal ideal of R containing the irreducible polynomial f. Moreover, we generalize these results for the universal module of nth order derivations.