Proof.
Parts (2) and (3) are equivalent by Cohomology, Lemma 20.50.5. If (1) is true, then for E perfect the derived tensor product E \otimes ^\mathbf {L} K is in D^ b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^ n_ k}) and we see that (2) holds by Derived Categories of Schemes, Lemma 36.11.3. It is clear that (2) implies (4) as \mathcal{O}_{\mathbf{P}^ n_ k}(d) can be viewed as a perfect object of the derived category of \mathbf{P}^ n_ k. Thus it suffices to prove that (4) implies (1).
Assume (4). Let R be as in Lemma 57.5.1. Let P = \bigoplus _{d = 0, \ldots , n} \mathcal{O}_{\mathbf{P}^ n_ k}(-d). Recall that R = \text{End}_{\mathbf{P}^ n_ k}(P) whereas all other self-Exts of P are zero and that P determines an equivalence - \otimes ^\mathbf {L} P : D(R) \to D_\mathit{QCoh}(\mathcal{O}_{\mathbf{P}^ n_ k}) by Derived Categories of Schemes, Lemma 36.20.1. Say K corresponds to L in D(R). Then
\begin{align*} H^ i(L) & = \mathop{\mathrm{Ext}}\nolimits ^ i_{D(R)}(R, L) \\ & = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathbf{P}^ n_ k}(P, K) \\ & = H^ i(\mathbf{P}^ n_ k, K \otimes P^\vee ) \\ & = \bigoplus \nolimits _{d = 0, \ldots , n} H^ i(\mathbf{P}^ n_ k, K \otimes \mathcal{O}(d)) \end{align*}
by Differential Graded Algebra, Lemma 22.35.4 (and the fact that - \otimes ^\mathbf {L} P is an equivalence) and Cohomology, Lemma 20.50.5. Thus our assumption (4) implies that L satisfies condition (2) of Lemma 57.5.1 and hence is a compact object of D(R). Therefore K is a compact object of D_\mathit{QCoh}(\mathcal{O}_{\mathbf{P}^ n_ k}). Thus K is perfect by Derived Categories of Schemes, Proposition 36.17.1. Since D_{perf}(\mathcal{O}_{\mathbf{P}^ n_ k}) = D^ b_{\textit{Coh}}(\mathcal{O}_{\mathbf{P}^ n_ k}) by Derived Categories of Schemes, Lemma 36.11.8 we conclude (1) holds.
\square
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