The chain rule of differentiation in superspace (Q1081908)

For functions on superspace, the superdifferential is either an ordinary real differential, which is a module homomorphism for the module structure over the even part of the exterior algebra, or one takes the Jacobi matrix of the formal partial derivatives when the function is expanded as a polynomial in the odd variables. But then the chain rule does not hold in general, since a nilpotent element may induce the zero action on some module. In this paper the authors propose a way to repair the chain rule for Jacobi matrices of formal partial derivatives. They define graded ideals of singular functions which are stable under formal differentiation and integration and they show that the deviation from the chain rule for formal partial derivatives falls into the singular functions. A simpler remedy is just to consider the associated module homomorphism as differential, then the chain rule holds.