[Beilinson]
Lemma 36.20.1. Let $A$ be a ring. Let $X = \mathbf{P}^ n_ A = \text{Proj}(S)$ where $S = A[X_0, \ldots , X_ n]$. With $P$ as in (36.20.0.1) and $R$ as in (36.20.0.2) the functor is an $A$-linear equivalence of triangulated categories sending $R$ to $P$.
Proof.
To prove that our functor is fully faithful it suffices to prove that $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(P, P)$ is zero for $i \not= 0$ and equal to $R$ for $i = 0$, see Differential Graded Algebra, Lemma 22.35.6. As in the proof of Lemma 36.13.5 we see that
By the computation of cohomology of projective space (Cohomology of Schemes, Lemma 30.8.1) we find that these $\mathop{\mathrm{Ext}}\nolimits $-groups are zero unless $i = 0$. For $i = 0$ we recover $R$ because this is how we defined $R$ in (36.20.0.2). By Differential Graded Algebra, Lemma 22.35.5 our functor has a right adjoint, namely $R\mathop{\mathrm{Hom}}\nolimits (P, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(R)$. Since $P$ is a generator for $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.16.3 we see that the kernel of $R\mathop{\mathrm{Hom}}\nolimits (P, -)$ is zero. Hence our functor is an equivalence of triangulated categories by Derived Categories, Lemma 13.7.2.
$\square$
\[ - \otimes _ R^\mathbf {L} P : D(R) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X) \]
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ X(P, P) = H^ i(X, P^\wedge \otimes P) = \bigoplus \nolimits _{0 \leq a, b \leq n} H^ i(X, \mathcal{O}_ X(a - b)) \]
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