Lemma 22.20.2. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then

for all acyclic differential graded $A$-modules $N$.

Lemma 22.20.2. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(P, N) = 0 \]

for all acyclic differential graded $A$-modules $N$.

**Proof.**
We will use that $K(\text{Mod}_{(A, \text{d})})$ is a triangulated category (Proposition 22.10.3). Let $F_\bullet $ be a filtration on $P$ as in property (P). The short exact sequence of Lemma 22.20.1 produces a distinguished triangle. Hence by Derived Categories, Lemma 13.4.2 it suffices to show that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(F_ iP, N) = 0 \]

for all acyclic differential graded $A$-modules $N$ and all $i$. Each of the differential graded modules $F_ iP$ has a finite filtration by admissible monomorphisms, whose graded pieces are direct sums of shifts $A[k]$. Thus it suffices to prove that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(A[k], N) = 0 \]

for all acyclic differential graded $A$-modules $N$ and all $k$. This follows from Lemma 22.16.2. $\square$

Your email address will not be published.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.

## Comments (0)

There are also: