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
of complexes of $R$-modules.
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
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
The element $\alpha ^{p, q}$ is a finite sum $\alpha ^{p, q} = \sum \beta ^ p_ i \otimes \gamma ^ q_ i$ with
and
The map is given by sending $\alpha $ to $\delta = (\delta ^{r, v})$ with
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
It follows that the rules $\alpha \mapsto \delta $ is compatible with differentials and the lemma is proved. $\square$
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