The Stacks project

Lemma 15.70.4. Let $R$ be a ring. Given complexes $K^\bullet , L^\bullet $ of $R$-modules there is a canonical morphism

\[ K^\bullet \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^\bullet (L^\bullet , \text{Tot}(K^\bullet \otimes _ R L^\bullet )) \]

of complexes of $R$-modules functorial in both complexes.

Proof. This is a special case of Lemma 15.70.3 but we will also construct it directly here. Let $\alpha $ be an element of degree $n$ of the right hand side. Thus

\[ \alpha = (\alpha ^{p, q}) \in \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, \text{Tot}^ p(K^\bullet \otimes _ R L^\bullet )) \]

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

\[ \alpha ^{p, q} = (\alpha ^{r, s, q}) \in \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, \bigoplus \nolimits _{r + s + q = n} K^ r \otimes _ R L^ s) \]

where we think of $\alpha ^{r, s, q}$ as a family of maps such that for every $x \in L^{-q}$ only a finite number of $\alpha ^{r, s, q}(x)$ are nonzero. By our sign rules we get

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

Now an element $\beta \in K^ n$ we send to $\alpha $ with $\alpha ^{n, -q, q} = \beta \otimes \text{id}_{L^{-q}}$ and $\alpha ^{r, s, q} = 0$ if $r \not= n$. This is indeed an element as above, as for fixed $q$ there is only one nonzero $\alpha ^{r, s, q}$. The description of the differential shows this is compatible with differentials. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

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 0A62. Beware of the difference between the letter 'O' and the digit '0'.