The Stacks project

Lemma 22.4.2. Let $(A, d)$ be a differential graded algebra. The category $\text{Mod}_{(A, \text{d})}$ is abelian and has arbitrary limits and colimits.

Proof. Kernels and cokernels commute with taking underlying $A$-modules. Similarly for direct sums and colimits. In other words, these operations in $\text{Mod}_{(A, \text{d})}$ commute with the forgetful functor to the category of $A$-modules. This is not the case for products and limits. Namely, if $N_ i$, $i \in I$ is a family of differential graded $A$-modules, then the product $\prod N_ i$ in $\text{Mod}_{(A, \text{d})}$ is given by setting $(\prod N_ i)^ n = \prod N_ i^ n$ and $\prod N_ i = \bigoplus _ n (\prod N_ i)^ n$. Thus we see that the product does commute with the forgetful functor to the category of graded $A$-modules. A category with products and equalizers has limits, see Categories, Lemma 4.14.11. $\square$

Comments (0)

There are also:

  • 5 comment(s) on Section 22.4: Differential graded modules

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Your email address will not be published.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 09JJ. Beware of the difference between the letter 'O' and the digit '0'.