It is well known that given a differential module $E$ with a differential d we can measure the non-exactness of this differential module by its homologies which are based on the key relation $d^2 =0$. This relation is a basis for several important structures in modern mathematics and theoretical physics to point out only two of them which are the theory of de Rham cohomologies on smooth manifolds and the apparatus of BRST-quantization in gauge field theories.
Viktor Abramov. Olga Liivapuu. "$N$-Complex, Graded $q$-Differential Algebra and $N$-Connection on Modules." J. Gen. Lie Theory Appl. 9 (S1) 1 - 11, 2015. https://doi.org/10.4172/1736-4337.S1-006