The Stacks project

[Beilinson]

Lemma 36.19.1. 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 (36.19.0.1) and $R$ as in (36.19.0.2) 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 $P$.

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 36.17.2 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 (36.19.0.2). 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 36.15.3 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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