nLab mixed complex

Idea

A mixed complex is both a chain complex and a cochain complex in a compatible way. Equivalently, it is a dg-module over the dg-algebra $k[B]/(B^2)$, where $k$ is the base ring, $B$ is of degree $-1$, and $dB = 0$. Mixed complexes were introduced in the study of cyclic homology.

References

• Christian Kassel, Cyclic homology, comodules and mixed complexes, J. Alg. 107 (1987), 195–216.