Lemma 15.71.3. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet , M^\bullet$ of $R$-modules there is a canonical morphism

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

of complexes of $R$-modules.

Proof. Via the discussion in Remark 15.71.2 the existence of such a canonical map follows from Categories, Remark 4.43.12. We also give a direct construction.

An element $\alpha$ of degree $n$ of the left hand side is

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

The element $\alpha ^{p, q}$ is a finite sum $\alpha ^{p, q} = \sum \beta ^ p_ i \otimes \gamma ^ q_ i$ with

$\beta ^ p_ i = (\beta ^{r, s}_ i) \in \prod \nolimits _{r + s = p} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-s}, M^ r)$

and

$\gamma ^ q_ i = (\gamma ^{u, v}_ i) \in \prod \nolimits _{u + v = q} \mathop{\mathrm{Hom}}\nolimits _ R(K^{-v}, L^ u)$

The map is given by sending $\alpha$ to $\delta = (\delta ^{r, v})$ with

$\delta ^{r, v} = \sum \nolimits _{i, s} \beta ^{r, s}_ i \circ \gamma ^{-s, v}_ i \in \mathop{\mathrm{Hom}}\nolimits _ R(K^{-v}, M^ r)$

For given $r + v = n$ this sum is finite as there are only finitely many nonzero $\alpha ^{p, q}$, hence only finitely many nonzero $\beta ^ p_ i$ and $\gamma ^ q_ i$. By our sign rules we have

\begin{align*} \text{d}(\alpha ^{p, q}) & = \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , M^\bullet )}(\alpha ^{p, q}) + (-1)^ p \text{d}_{\mathop{\mathrm{Hom}}\nolimits ^\bullet (K^\bullet , L^\bullet )}(\alpha ^{p, q}) \\ & = \sum \Big( \text{d}_ M \circ \beta ^ p_ i \circ \gamma ^ q_ i - (-1)^ p \beta ^ p_ i \circ \text{d}_ L \circ \gamma ^ q_ i \Big) \\ & \quad + (-1)^ p \sum \Big( \beta ^ p_ i \circ \text{d}_ L \circ \gamma ^ q_ i - (-1)^ q \beta ^ p_ i \circ \gamma ^ q_ i \circ \text{d}_ K \Big) \\ & = \sum \Big( \text{d}_ M \circ \beta ^ p_ i \circ \gamma ^ q_ i -(-1)^ n \beta ^ p_ i \circ \gamma ^ q_ i \circ \text{d}_ K \Big) \end{align*}

It follows that the rules $\alpha \mapsto \delta$ is compatible with differentials and the lemma is proved. $\square$

Comment #7132 by Hao Peng on

Just to point out this seems to be a direct consequence of tag 0A5Y.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A8I. Beware of the difference between the letter 'O' and the digit '0'.