36.20 An example equivalence

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

36.20.0.1
$$\label{perfect-equation-generator-Pn} P = \mathcal{O}_ X \oplus \mathcal{O}_ X(-1) \oplus \ldots \oplus \mathcal{O}_ X(-n)$$

is a perfect generator of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Consider the (noncommutative) $A$-algebra

36.20.0.2
$$\label{perfect-equation-algebra-for-Pn} R = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(P, P) = \left( \begin{matrix} S_0 & S_1 & S_2 & \ldots & \ldots \\ 0 & S_0 & S_1 & \ldots & \ldots \\ 0 & 0 & S_0 & \ldots & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & \ldots & S_0 \end{matrix} \right)$$

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

$R = \mathop{\mathrm{Hom}}\nolimits _{\text{Comp}^{dg}(\mathcal{O}_ X)}(P, P)$

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

$- \otimes _ R^\mathbf {L} P : D(R) \longrightarrow D(\mathcal{O}_ X),$

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

$- \otimes _ R^\mathbf {L} P : D(R) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X)$

is an $A$-linear equivalence of triangulated categories sending $R$ to $P$.

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

$\mathop{\mathrm{Ext}}\nolimits ^ i_ X(P, P) = H^ i(X, P^\wedge \otimes P) = \bigoplus \nolimits _{0 \leq a, b \leq n} H^ i(X, \mathcal{O}_ X(a - b))$

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. $\square$

Comments (2)

Comment #2720 by Zhang on

Reference: Beilinson (1978) "Coherent sheaves on P^n and problems in linear algebra".

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