The technique of introducing singularities of some form to a bordism theory proposed in [7] by Sullivan and developed in [1] has proved itself to be a very efficient method of constructing new homology and cohomology theories with predetermined properties (see, for example [3]–[6]). We propose a generalization of this method taking as our starting point a complex of free U*-modules endowed with an additional ‘geometric’ structure. The Sullivan—Baas construction of killing a regular sequence x1, x2, … of elements of U* fits in our framework as a special case of Koszul resolution of U*/(x1, x2, …) with some canonical structure. Also, when specialized to the U*-complex of the form U* ⊗Z C for a ℤ-complex of length 3, our construction turns out to be equivalent to the ‘bordisms with coefficients’ described in [2].