chain complex

Last updated January 27, 2022

Definition

A sequence of modules such that $d^2=0$, i.e. the image of the previous map is contained in the kernel of the next map.

• (Projective) A projective chain complex (augmented) over $M$ is a chain complex $$P_\ast\to M\to 0$$ such that the $P_\ast$ are projective modules (see inj proj modules).
  • Note that this is more general than a projective resolution (see inj proj resolutions), as we are not requiring that the sequence is exact.