Differential of a chain complex ($\partial$)

The chain differential of a homological chain complex is a family of group or module homomorphisms ${\partial }_n:C_n\to C_{n-1}$ satisfying ${\partial }_{n-1}{\partial }_n=0$. The identity implies $\mathrm{im}{\partial }_{n+1}\subset \ker {\partial }_n$, so the quotient $H_n=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$ is defined. In a graded complex over $\mathbb{Z}[U]$, the chain differential can have degree $-1$ while $U$ has degree $-2$, with $\partial (Ux)=U\partial x$.