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
Comments (2)
Comment #7132 by Hao Peng on
Comment #7290 by Johan on