# Hom cochain complex of two chain complexes

Does anybody know of a good reference (preferably online) where I can find a good, rigorous description of $Hom_R(C_\bullet,D_\bullet)_\bullet$, which is a cochain complex where the module is the nth degree is given as $\Pi_{i-j=n}Hom_R(C_i,D_j)$, where $C_\bullet$ and $D_\bullet$ are chain complexes? I would give what the coboundary maps are, but that is the main piece of information I am trying to find a definition of, as well as some other things about it. Thanks!

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The boundary in the complex $\hom(C_\bullet, D_\bullet)$ is simply the map $$\delta:\prod_{i-j=n}Hom_R(C_j,D_i) \to \prod_{i-j=n-1}Hom_R(C_j,D_i)$$ such that for each $f\in Hom_R(C_i,D_j)$ we have $$\delta(f)=f\circ d_C-d_D\circ f.$$ (You do have to figure out in which factor of the codomain of $\delta$ each of these two summands is; notice also that $0$-cycles are precisely the morphisms of complexes, and you should try to see what exactly does it mean for two such $0$-cocycles to be homologous)
I have some difficulty with that definition. If $f:C_i\to D_j$ then when how can we compose $f$ with $d_C$, since after we apply $d_C$ we are in $C_{i-1}$? – Jon Beardsley Mar 11 '11 at 2:45
Well, $C$ has a lot of $d_C$s... use the one that has $C_i$ as codomain! – Mariano Suárez-Alvarez Mar 11 '11 at 2:48
Okay, this is fine. What I was failing to realize was that $f$ is actually an infinite collection of maps, not one map for one $i$, so when we talk about $d_Cf$ we mean applying the relevant coordinate of $f$ at the right spot. Thankyou! – Jon Beardsley Mar 11 '11 at 3:41