# reference query – tensor and hom complex of graduated module complexes

Yes $$R$$ is a commutative ring, $$^ cdot, Y ^ cdot$$ are complexes of $$R$$ modules, it is natural to define $$X ^ cdot otimes ^ cdot Y ^ cdot$$ by the nth component $$coprod_ {i + j = n} X ^ i otimes_R Y ^ j$$ with differential:$$d (x otimes y) = dx otimes y + (- 1) ^ { mid x mid} otimes$$and there is a deputy $$(X ^ cdot otimes ^ cdot-, Hom ^ cdot (X ^ cdot, -)$$ between DG category $$C_ {d}} (R)$$ and $$C_ {d}} (R)$$.

I want to find the graduated version of the tensor and hom complexes of graduated module complexes.

Yes $$A$$ is a commutative graduated ring. $$A-Gr$$ graduation category $$A$$ modules with homomorphism homogeneous morphism $$A$$-modules.Let $$M, N$$ be ranked two $$A-$$modules. It is also natural to define $$M otimes ^ * N = coprod _n (M ototimes ^ * N) ^ n$$ with n-th component is generated by $$x otimes y$$ such as $$mid x mid + mid y mid = n$$.To define $$^ * Hom (M, N) = coprod_n Hom_ {A-Gr} (M, N)[n]$$.
If to consider $$A = A ^ 0$$,then $$Hom_ {A-Gr} (M, N) = prod_nHom_A (M ^ n, N ^ n)$$, direct calculation impossible $$(M otimes ^ * -, ^ * Hom (M, -))$$ is an assistant couple.So I think that in general, it is not assistant, right?

Consider complexes of $$A$$ modules $$X ^ cdot, Y ^ cdot$$How to define $$X ^ cdot otimes ^ cdot Y ^ cdot$$ and $$Hom ^ cdot (X ^ cdot, Y ^ cdot)$$?

Similarly, define $$(X ^ cdot otimes ^ cdot Y cdot) ^ n = coprod_ {i + j = n} X ^ i otimes ^ * Y ^ j$$but how to define the differential? Yes $$x in (X ^ i) _j$$,or $$X ^ i$$ is a noted module. There are two different ways of defining $$d (x otimes y) = dx otimes y + (- 1) ^ {i} x otimes dy$$ or replaced $$(- 1) ^ i$$ by $$(- 1) ^ {i + j}$$Both cases can make it complex.It seems that $$(- 1) ^ {i + j}$$ Maybe better. What is the standard way? that is to say what is the degree of $$x$$? How to define the graduated version differential of the hom complex?

Are there good literatures on the class of graduated module complexes? With homological details.

Thank you in advance.