Flagged Perturbations and Anchored Resolutions

In this paper, we take advantage of a reinterpretation of differential modules admitting a flag structure as a special class of perturbations of complexes. We are thus able to leverage the machinery of homological perturbation theory to prove strong statements on the homological theory of differential modules admitting additional auxiliary gradings and having infinite homological dimension. One of the main takeaways of our results is that the category of differential modules is much more similar than expected to the category of chain complexes, and from the K-theoretic perspective such objects are largely indistinguishable. This intuition is made precise through the construction of so-called anchored resolutions, which are a distinguished class of projective flag resolutions that possess remarkably well-behaved uniqueness properties in the (flag-preserving) homotopy category. We apply this theory to prove an analogue of the Total Rank Conjecture for differential modules admitting a ZZ/2-grading in a large number of cases.