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
Comments (0)