The main object of this paper is the comparison of several theorems in homological algebra. The theorems are found in [1] [V. K. A. M. Gugenheim and H. J. Munkholm, J. Pure Appl. Algebra 4, 9-29 (1974; Zbl 0358.18015)], [2] [V. K. A. M. Gugenheim and J. Stasheff, Bull. Soc. Math. Belg., Ser. A 38, 237-246 (1986; Zbl 0639.55008)] and [3] [V. K. A. M. Gugenheim and J. P. May, Mem. Am. Math. Soc. 142 (1974; Zbl 0292.55019)]. In particular, [1] is shown to be a special case of [2]. It is furthermore shown how special cases of the “distinguished resolution” of [3] can be constructed using the algorithms for “twisting cochains” discussed in this paper and hence they give an alternate method for constructing the free resolutions over the group ring of certain two-stage nilpotent groups given earlier by the second author [Bull. Soc. Math. Belg., Ser. A 38, 247-264 (1986; Zbl 0632.55010) and Computers in geometry and topology, Proc. Conf., Chicago/Ill. 1986, Lect. Notes Pure Appl. Math. 114, 189-209 (1989)].