Taylor and Lyubeznik resolutions via Gröbner bases
Journal of Symbolic Computation
Computer algebra handbook
Comparison maps for relatively free resolutions
CASC'06 Proceedings of the 9th international conference on Computer Algebra in Scientific Computing
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Perturbation theory is a useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is ''the basic perturbation lemma'' which allows the transfer of structure between differential graded modules M and N of the same homotopy type when the differential in one has been changed. Under certain circumstances, the theory can be applied to obtain resolutions over algebras which are ''perturbations'' of algebras over which resolutions are known. For example, if D is any finitely generated torsion-free nilpotent group then the integral group ring A=Z(D) may be considered as a perturbation of the ring of Laurent polynomials. The theory may then be applied to a well-known resolution over the Laurent polynomials to obtain a resolution over the group ring. More general classes of algebras are discussed in ''Homological Perturbation theory, Hochschild homology and formal groups'' by the author (to appear). There is a trade-off between the size of the resolutions which arise from the perturbation method and the complexity of the new differential. In order to keep the modules relatively small, there is a considerable increase in the algebraic complexity of the resulting differentials. In order to study such complexes systematically, examples are needed, To facilitate such study, the Scratchpad system was used to set up and perform the necessary calculations. Because of the way Scratchpad is organized, this could be done in a way that minimizes programming effort and provides the natural mathematical environment for such calculations. In this paper, we discuss some of the general theory behind homological perturbation theory, give an idea of what is needed to make calculations within that theory in Scratchpad, and calculate a resolution of the integers over the integral group ring of the 4 x 4 upper triangular matrices with ones along the diagonal.