The Stacks project

Example 12.18.2. Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ be additive categories. Suppose that

\[ \otimes : \mathcal{A} \times \mathcal{B} \longrightarrow \mathcal{C}, \quad (X, Y) \longmapsto X \otimes Y \]

is a functor which is bilinear on morphisms, see Categories, Definition 4.2.20 for the definition of $\mathcal{A} \times \mathcal{B}$. Given complexes $X^\bullet $ of $\mathcal{A}$ and $Y^\bullet $ of $\mathcal{B}$ we obtain a double complex

\[ K^{\bullet , \bullet } = X^\bullet \otimes Y^\bullet \]

in $\mathcal{C}$. Here the first differential $K^{p, q} \to K^{p + 1, q}$ is the morphism $X^ p \otimes Y^ q \to X^{p + 1} \otimes Y^ q$ induced by the morphism $X^ p \to X^{p + 1}$ and the identity on $Y^ q$. Similarly for the second differential.


Comments (2)

Comment #3069 by anon on

Typo: Please delete "a" in "a complexes".


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Your email address will not be published.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 0A5J. Beware of the difference between the letter 'O' and the digit '0'.