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Let $A$ and $B$ be differential graded algebras over a field $k$ and $D(A)$ and $D(B)$ be the derived categories of differential graded modules.

Let $N$ be an $A$-$B$ bimodule. Then $(-)\otimes^{\mathbf{L}}_A N$ gives a triangulated functor $D(A)\to D(B)$. It is well-known that we have the follow result. See Stack Project Tag 09R9, Lemma 22.24.6 .

Assume

  1. $N$ defines a compact object of $D(B)$;
  2. The map $H^i(A)\to \text{Hom}_{D(B)}(N,N[i])$ is an isomorphism for any $i\in \mathbb{Z}$.

Then $(-)\otimes^{\mathbf{L}}_A N$ is a fully faithful functor.

I would like to know whether the inverse statement is true. In more details, if we assume that $N$ defines a compact object of $D(B)$ and $(-)\otimes^{\mathbf{L}}_A N: D(A)\to D(B)$ is fully faithful, is it true that $H^i(A)\to \text{Hom}_{D(B)}(N,N[i])$ is an isomorphism for any $i\in \mathbb{Z}$?

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    $\begingroup$ Isn't $H^i(A)=\mathrm{Hom}_{D(A)}(A,A[i])$, so the claim follows immediately from full faithfulness (without even the compactness of $N$)? $\endgroup$
    – Denis Nardin
    Jul 27, 2016 at 14:12
  • $\begingroup$ @DenisNardin Oh yes that's so obvious. Thank you. $\endgroup$
    – Zhaoting Wei
    Jul 27, 2016 at 14:19

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