In Section 36.16 we proved that the derived category of projective space $\mathbf{P}^ n_ A$ over a ring $A$ is generated by a vector bundle, in fact a direct sum of shifts of the structure sheaf. In this section we prove this determines an equivalence of $D_\mathit{QCoh}(\mathcal{O}_{\mathbf{P}^ n_ A})$ with the derived category of an $A$-algebra.
Before we can state the result we need some notation. Let $A$ be a ring. Let $X = \mathbf{P}^ n_ A = \text{Proj}(S)$ where $S = A[X_0, \ldots , X_ n]$. By Lemma 36.16.3 we know that
is a perfect generator of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Consider the (noncommutative) $A$-algebra
with obvious multiplication and addition. If we view $P$ as a complex of $\mathcal{O}_ X$-modules in the usual way (i.e., with $P$ in degree $0$ and zero in every other degree), then we have
where on the right hand side we view $R$ as a differential graded algebra over $A$ with zero differential (i.e., with $R$ in degree $0$ and zero in every other degree). According to the discussion in Differential Graded Algebra, Section 22.35 we obtain a derived functor
see especially Differential Graded Algebra, Lemma 22.35.3. By Lemma 36.18.1 we see that the essential image of this functor is contained in $D_\mathit{QCoh}(\mathcal{O}_ X)$.
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 $P$.
\[ - \otimes _ R^\mathbf {L} P : D(R) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X) \]
In words: the derived category of quasi-coherent modules on projective space is equivalent to the derived category of modules over a (noncommutative) algebra. This property of projective space appears to be quite unusual among all projective schemes over $A$.
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 36.18.2 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. $\square$
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