Lemma 15.67.1. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet$ of $R$-modules there is a canonical isomorphism

$\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )) = \mathop{\mathrm{Hom}}\nolimits ^\bullet (\text{Tot}(K^\bullet \otimes _ R L^\bullet ), M^\bullet )$

of complexes of $R$-modules.

Proof. Let $\alpha$ be an element of degree $n$ on the left hand side. Thus

$\alpha = (\alpha ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits ^ p(L^\bullet , M^\bullet ))$

Each $\alpha ^{p, q}$ is an element

$\alpha ^{p, q} = (\alpha ^{r, s, q}) \in \prod \nolimits _{r + s + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r))$

If we make the identifications

15.67.1.1
$$\label{more-algebra-equation-identification} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q}, \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r)) = \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q} \otimes _ R L^{-s}, M^ r)$$

then by our sign rules we get

\begin{align*} \text{d}(\alpha ^{r, s, q}) & = \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )} \circ \alpha ^{r, s, q} - (-1)^ n \alpha ^{r, s, q} \circ \text{d}_ K \\ & = \text{d}_ M \circ \alpha ^{r, s, q} - (-1)^{r + s} \alpha ^{r, s, q} \circ \text{d}_ L - (-1)^{r + s + q} \alpha ^{r, s, q} \circ \text{d}_ K \end{align*}

On the other hand, if $\beta$ is an element of degree $n$ of the right hand side, then

$\beta = (\beta ^{r, s, q}) \in \prod \nolimits _{r + s + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-q} \otimes _ R L^{-s}, M^ r)$

and by our sign rule (Homology, Definition 12.22.3) we get

\begin{align*} \text{d}(\beta ^{r, s, q}) & = \text{d}_ M \circ \beta ^{r, s, q} - (-1)^ n \beta ^{r, s, q} \circ \text{d}_{\text{Tot}(K^\bullet \otimes L^\bullet )} \\ & = \text{d}_ M \circ \beta ^{r, s, q} - (-1)^{r + s + q} \left( \beta ^{r, s, q} \circ \text{d}_ K + (-1)^{-q} \beta ^{r, s, q} \circ \text{d}_ L \right) \end{align*}

Thus we see that the map induced by the identifications (15.67.1.1) indeed is a morphism of complexes. $\square$

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