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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