Tensor product of chain complexes ($(C\otimes C^{\prime }{)}_k={⨁}_{p+q=k}{C}_p\otimes C_q^{\prime }$)

The tensor product of homologically graded chain complexes has differential $D(x_p\otimes y)=d{x}_p\otimes y+(-1{)}^p{x}_p\otimes dy$. The sign makes the two mixed terms in $D^2$ cancel. Total degree is the sum of the two degrees.

Interchanging homogeneous objects of degrees $p$ and $q$ introduces $(-1{)}^{pq}$. In particular, moving a degree-minus-one differential past a degree-$p$ factor introduces $(-1{)}^p$. This convention produces the differential on the tensor product of chain complexes.