The Stacks project

Lemma 22.29.1. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Then $M \mapsto M \otimes _ A N$ defines a functor

\[ - \otimes _ A N : \text{Mod}^{dg}_{(A, \text{d})} \longrightarrow \text{Mod}^{dg}_{(B, \text{d})} \]

of differential graded categories. This functor induces functors

\[ \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(B, \text{d})} \quad \text{and}\quad K(\text{Mod}_{(A, \text{d})}) \to K(\text{Mod}_{(B, \text{d})}) \]

by an application of Lemma 22.26.5.

Proof. Above we have seen how the construction defines a functor of underlying graded categories. Thus it suffices to show that the construction is compatible with differentials. Let $M$ and $M'$ be differential graded $A$-modules and let $f : M \to M'$ be an $A$-module homomorphism which is homogeneous of degree $n$. Then we have

\[ \text{d}(f) = \text{d}_{M'} \circ f - (-1)^ n f \circ \text{d}_ M \]

On the other hand, we have

\[ \text{d}(f \otimes \text{id}_ N) = \text{d}_{M' \otimes _ A N} \circ (f \otimes \text{id}_ N) - (-1)^ n (f \otimes \text{id}_ N) \circ \text{d}_{M \otimes _ A N} \]

Applying this to an element $x \otimes y$ with $x \in M$ and $y \in N$ homogeneous we get

\begin{align*} \text{d}(f \otimes \text{id}_ N)(x \otimes y) = & \text{d}_{M'}(f(x)) \otimes y + (-1)^{n + \deg (x)}f(x) \otimes \text{d}_ N(y) \\ & - (-1)^ n f(\text{d}_ M(x)) \otimes y - (-1)^{n + \deg (x)}f(x) \otimes \text{d}_ N(y) \\ = & \text{d}(f) (x \otimes y) \end{align*}

Thus we see that $\text{d}(f) \otimes \text{id}_ N = \text{d}(f \otimes \text{id}_ N)$ and the proof is complete. $\square$

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Your email address will not be published.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