= Differential of a chain complex
{title2=$\partial$}
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 $\operatorname{im}\partial_{n+1}\subset\ker\partial_n$, so the quotient $H_n=\ker\partial_n/\operatorname{im}\partial_{n+1}$ is defined. In a graded complex over $\mathbb Z[U]$, the <chain differential> can have degree $-1$ while $U$ has degree $-2$, with $\partial(Ux)=U\partial x$.
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