Graded Hom complex of chain complexes ($M_j(C,C^{\prime })={\prod }_p{\mathrm{Hom}}_R(C_p,C_{p+j}^{\prime })$)

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

For finite free total chain complexes, evaluation identifies the underlying graded tensor product $X\otimes C^{\prime }$ with the graded Hom complex of chain complexes, where $X$ is the reversed dual chain complex. Under the precomposition-first Hom differential, conjugation by the sign $(-1{)}^{j(j-1)/2}$ makes this a chain isomorphism. The identity $\rho (j)-\rho (j-1)=j-1$ works for every integer degree. If the grading is unbounded and only degreewise finite, products on the Hom side need not equal direct sums on the tensor side.