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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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