The Stacks project

36.7 The coherator for Noetherian schemes

In the case of Noetherian schemes we can use the following lemma.

Lemma 36.7.1. Let $X$ be a Noetherian scheme. Let $\mathcal{J}$ be an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$. Then $\mathcal{J}$ is a flasque sheaf of $\mathcal{O}_ X$-modules.

Proof. Let $U \subset X$ be an open subset and let $s \in \mathcal{J}(U)$ be a section. Let $\mathcal{I} \subset X$ be the quasi-coherent sheaf of ideals defining the reduced induced scheme structure on $X \setminus U$ (see Schemes, Definition 26.12.5). By Cohomology of Schemes, Lemma 30.10.4 the section $s$ corresponds to a map $\sigma : \mathcal{I}^ n \to \mathcal{J}$ for some $n$. As $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$ we can extend $\sigma $ to a map $\tilde s : \mathcal{O}_ X \to \mathcal{J}$. Then $\tilde s$ corresponds to a global section of $\mathcal{J}$ restricting to $s$. $\square$

Lemma 36.7.2. Let $f : X \to Y$ be a morphism of Noetherian schemes. Then $f_*$ on quasi-coherent sheaves has a right derived extension $\Phi : D(\mathit{QCoh}(\mathcal{O}_ X)) \to D(\mathit{QCoh}(\mathcal{O}_ Y))$ such that the diagram

\[ \xymatrix{ D(\mathit{QCoh}(\mathcal{O}_ X)) \ar[d]_{\Phi } \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ X) \ar[d]^{Rf_*} \\ D(\mathit{QCoh}(\mathcal{O}_ Y)) \ar[r] & D_\mathit{QCoh}(\mathcal{O}_ Y) } \]

commutes.

Proof. Since $X$ and $Y$ are Noetherian schemes the morphism is quasi-compact and quasi-separated (see Properties, Lemma 28.5.4 and Schemes, Remark 26.21.18). Thus $f_*$ preserve quasi-coherence, see Schemes, Lemma 26.24.1. Next, let $K$ be an object of $D(\mathit{QCoh}(\mathcal{O}_ X))$. Since $\mathit{QCoh}(\mathcal{O}_ X)$ is a Grothendieck abelian category (Properties, Proposition 28.23.4), we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet $ such that each $\mathcal{I}^ n$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, see Injectives, Theorem 19.12.6. Thus we see that the functor $\Phi $ is defined by setting

\[ \Phi (K) = f_*\mathcal{I}^\bullet \]

where the right hand side is viewed as an object of $D(\mathit{QCoh}(\mathcal{O}_ Y))$. To finish the proof of the lemma it suffices to show that the canonical map

\[ f_*\mathcal{I}^\bullet \longrightarrow Rf_*\mathcal{I}^\bullet \]

is an isomorphism in $D(\mathcal{O}_ Y)$. To see this it suffices to prove the map induces an isomorphism on cohomology sheaves. Pick any $m \in \mathbf{Z}$. Let $N = N(X, Y, f)$ be as in Lemma 36.4.1. Consider the short exact sequence

\[ 0 \to \sigma _{\geq m - N - 1}\mathcal{I}^\bullet \to \mathcal{I}^\bullet \to \sigma _{\leq m - N - 2}\mathcal{I}^\bullet \to 0 \]

of complexes of quasi-coherent sheaves on $X$. By Lemma 36.4.1 we see that the cohomology sheaves of $Rf_*\sigma _{\leq m - N - 2}\mathcal{I}^\bullet $ are zero in degrees $\geq m - 1$. Thus we see that $R^ mf_*\mathcal{I}^\bullet $ is isomorphic to $R^ mf_*\sigma _{\geq m - N - 1}\mathcal{I}^\bullet $. In other words, we may assume that $\mathcal{I}^\bullet $ is a bounded below complex of injective objects of $\mathit{QCoh}(\mathcal{O}_ X)$. This follows from Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) via Cohomology, Lemma 20.12.5 and Lemma 36.7.1. $\square$

Proposition 36.7.3. Let $X$ be a Noetherian scheme. Then the functor (36.3.0.1)

\[ D(\mathit{QCoh}(\mathcal{O}_ X)) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ X) \]

is an equivalence with quasi-inverse given by $RQ_ X$.

Proof. This follows from Lemma 36.6.5 and Lemma 36.7.2. $\square$


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