A complex

is a chain of abelian groups $C_k$ and homomorphisms $\partial_k$ such that $\partial_k\circ\partial_{k+1}=0$ for all $k$, that is, $\text{im }\partial_{k+1}\subset \text{ker }\partial_k$

Associated with the complex $C_\ast$ we define the cohomology groups

[Cohomological version]

A complex

is a chain of abelian groups $C^k$ and homomorphisms $\mathrm{d}^k$ such that $\mathrm{d}^k\circ\mathrm{d}^{k-1}=0$ for all $k$, that is, $\text{im }\mathrm{d}^{k-1}\subset \text{ker }\mathrm{d}^k$

Associated with the complex $C^\ast$ we define the cohomology groups