The Stacks project

Lemma 36.5.4. Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ induces an equivalence

\[ \Phi : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(f_*\mathcal{O}_ X) \]

whose composition with $D_\mathit{QCoh}(f_*\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. Recall that $Rf_*$ is computed on an object $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ by choosing a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X$-modules representing $K$ and taking $f_*\mathcal{I}^\bullet $. Thus we let $\Phi (K)$ be the complex $f_*\mathcal{I}^\bullet $ viewed as a complex of $f_*\mathcal{O}_ X$-modules. Denote $g : (X, \mathcal{O}_ X) \to (Y, f_*\mathcal{O}_ X)$ the obvious morphism of ringed spaces. Then $g$ is a flat morphism of ringed spaces (see below for a description of the stalks) and $\Phi $ is the restriction of $Rg_*$ to $D_\mathit{QCoh}(\mathcal{O}_ X)$. We claim that $Lg^*$ is a quasi-inverse. First, observe that $Lg^*$ sends $D_\mathit{QCoh}(f_*\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ X)$ because $g^*$ transforms quasi-coherent modules into quasi-coherent modules (Modules, Lemma 17.10.4). To finish the proof it suffices to show that the adjunction mappings

\[ Lg^*\Phi (K) = Lg^*Rg_*K \to K \quad \text{and}\quad M \to Rg_*Lg^*M = \Phi (Lg^*M) \]

are isomorphisms for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $M \in D_\mathit{QCoh}(f_*\mathcal{O}_ X)$. This is a local question, hence we may assume $Y$ and therefore $X$ are affine.

Assume $Y = \mathop{\mathrm{Spec}}(B)$ and $X = \mathop{\mathrm{Spec}}(A)$. Let $\mathfrak p = x \in \mathop{\mathrm{Spec}}(A) = X$ be a point mapping to $\mathfrak q = y \in \mathop{\mathrm{Spec}}(B) = Y$. Then $(f_*\mathcal{O}_ X)_ y = A_\mathfrak q$ and $\mathcal{O}_{X, x} = A_\mathfrak p$ hence $g$ is flat. Hence $g^*$ is exact and $H^ i(Lg^*M) = g^*H^ i(M)$ for any $M$ in $D(f_*\mathcal{O}_ X)$. For $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we see that

\[ H^ i(\Phi (K)) = H^ i(Rf_*K) = f_*H^ i(K) \]

by the vanishing of higher direct images (Cohomology of Schemes, Lemma 30.2.3) and Lemma 36.3.4 (small detail omitted). Thus it suffice to show that

\[ g^*g_*\mathcal{F} \to \mathcal{F} \quad \text{and}\quad \mathcal{G} \to g_*g^*\mathcal{F} \]

are isomorphisms where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module and $\mathcal{G}$ is a quasi-coherent $f_*\mathcal{O}_ X$-module. This follows from Morphisms, Lemma 29.11.6. $\square$


Comments (1)

Comment #8293 by Firmaprim on

There is a typo at the end. It should be .


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