The Stacks project

Lemma 22.33.7. With notation and assumptions as in Lemma 22.33.5. Assume

  1. $N$ defines a compact object of $D(B, \text{d})$, and

  2. the map $H^ k(A) \to \mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N[k])$ is an isomorphism for all $k \in \mathbf{Z}$.

Then the functor $-\otimes _ A^\mathbf {L} N$ is fully faithful.

Proof. Our functor has a left adjoint given by $R\mathop{\mathrm{Hom}}\nolimits (N, -)$ by Lemma 22.33.5. By Categories, Lemma 4.24.4 it suffices to show that for a differential graded $A$-module $M$ the map

\[ M \longrightarrow R\mathop{\mathrm{Hom}}\nolimits (N, M \otimes _ A^\mathbf {L} N) \]

is an isomorphism in $D(A, \text{d})$. For this it suffices to show that

\[ H^ n(M) \longrightarrow \text{Ext}^ n_{D(B, \text{d})}(N, M \otimes _ A^\mathbf {L} N) \]

is an isomorphism, see Lemma 22.31.4. Since $N$ is a compact object the right hand side commutes with direct sums. Thus by Remark 22.22.5 it suffices to prove this map is an isomorphism for $M = A[k]$. Since $(A[k] \otimes _ A^\mathbf {L} N) = N[k]$ by Remark 22.29.2, assumption (2) on $N$ is that the result holds for these. $\square$

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