The Stacks project

72.12 The coherator for Noetherian spaces

We need a little bit more about injective modules to treat the case of a Noetherian algebraic space.

Lemma 72.12.1. Let $S$ be a Noetherian affine scheme. Every injective object of $\mathit{QCoh}(\mathcal{O}_ S)$ is a filtered colimit $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of quasi-coherent sheaves of the form

\[ \mathcal{F}_ i = (Z_ i \to S)_*\mathcal{G}_ i \]

where $Z_ i$ is the spectrum of an Artinian ring and $\mathcal{G}_ i$ is a coherent module on $Z_ i$.

Proof. Let $S = \mathop{\mathrm{Spec}}(A)$. Let $\mathcal{J}$ be an injective object of $\mathit{QCoh}(\mathcal{O}_ S)$. Since $\mathit{QCoh}(\mathcal{O}_ S)$ is equivalent to the category of $A$-modules we see that $\mathcal{J}$ is equal to $\widetilde{J}$ for some injective $A$-module $J$. By Dualizing Complexes, Proposition 47.5.9 we can write $J = \bigoplus E_\alpha $ with $E_\alpha $ indecomposable and therefore isomorphic to the injective hull of a reside field at a point. Thus (because finite disjoint unions of Artinian schemes are Artinian) we may assume that $J$ is the injective hull of $\kappa (\mathfrak p)$ for some prime $\mathfrak p$ of $A$. Then $J = \bigcup J[\mathfrak p^ n]$ where $J[\mathfrak p^ n]$ is the injective hull of $\kappa (\mathfrak p)$ over $A_\mathfrak /\mathfrak p^ nA_\mathfrak p$, see Dualizing Complexes, Lemma 47.7.3. Thus $\widetilde{J}$ is the colimit of the sheaves $(Z_ n \to X)_*\mathcal{G}_ n$ where $Z_ n = \mathop{\mathrm{Spec}}(A_\mathfrak p/\mathfrak p^ nA_\mathfrak p)$ and $\mathfrak G_ n$ the coherent sheaf associated to the finite $A_\mathfrak /\mathfrak p^ nA_\mathfrak p$-module $J[\mathfrak p^ n]$. Finiteness follows from Dualizing Complexes, Lemma 47.6.1. $\square$

Lemma 72.12.2. Let $S$ be an affine scheme. Let $X$ be a Noetherian algebraic space over $S$. Every injective object of $\mathit{QCoh}(\mathcal{O}_ X)$ is a direct summand of a filtered colimit $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of quasi-coherent sheaves of the form

\[ \mathcal{F}_ i = (Z_ i \to X)_*\mathcal{G}_ i \]

where $Z_ i$ is the spectrum of an Artinian ring and $\mathcal{G}_ i$ is a coherent module on $Z_ i$.

Proof. Choose an affine scheme $U$ and a surjective étale morphism $j : U \to X$ (Properties of Spaces, Lemma 63.6.3). Then $U$ is a Noetherian affine scheme. Choose an injective object $\mathcal{J}'$ of $\mathit{QCoh}(\mathcal{O}_ U)$ such that there exists an injection $\mathcal{J}|_ U \to \mathcal{J}'$. Then

\[ \mathcal{J} \to j_*\mathcal{J}' \]

is an injective morphism in $\mathit{QCoh}(\mathcal{O}_ X)$, hence identifies $\mathcal{J}$ as a direct summand of $j_*\mathcal{J}'$. Thus the result follows from the corresponding result for $\mathcal{J}'$ proved in Lemma 72.12.1. $\square$

Lemma 72.12.3. Let $S$ be a scheme. Let $f : X \to Y$ be a flat, quasi-compact, and quasi-separated morphism of algebraic spaces over $S$. If $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, then $f_*\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. Since $f$ is quasi-compact and quasi-separated, the functor $f_*$ transforms quasi-coherent sheaves into quasi-coherent sheaves (Morphisms of Spaces, Lemma 64.11.2). The functor $f^*$ is a left adjoint to $f_*$ which transforms injections into injections. Hence the result follows from Homology, Lemma 12.29.1 $\square$

Lemma 72.12.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. If $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$, then

  1. $H^ p(U, \mathcal{J}|_ U) = 0$ for $p > 0$ and for every quasi-compact and quasi-separated algebraic space $U$ étale over $X$,

  2. for any morphism $f : X \to Y$ of algebraic spaces over $S$ with $Y$ quasi-separated we have $R^ pf_*\mathcal{J} = 0$ for $p > 0$.

Proof. Proof of (1). Write $\mathcal{J}$ as a direct summand of $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ with $\mathcal{F}_ i = (Z_ i \to X)_*\mathcal{G}_ i$ as in Lemma 72.12.2. It is clear that it suffices to prove the vanishing for $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$. Since pullback commutes with colimits and since $U$ is quasi-compact and quasi-separated, it suffices to prove $H^ p(U, \mathcal{F}_ i|_ U) = 0$ for $p > 0$, see Cohomology of Spaces, Lemma 66.5.1. Observe that $Z_ i \to X$ is an affine morphism, see Morphisms of Spaces, Lemma 64.20.12. Thus

\[ \mathcal{F}_ i|_ U = (Z_ i \times _ X U \to U)_*\mathcal{G}'_ i = R(Z_ i \times _ X U \to U)_*\mathcal{G}'_ i \]

where $\mathcal{G}'_ i$ is the pullback of $\mathcal{G}_ i$ to $Z_ i \times _ X U$, see Cohomology of Spaces, Lemma 66.11.1. Since $Z_ i \times _ X U$ is affine we conclude that $\mathcal{G}'_ i$ has no higher cohomology on $Z_ i \times _ X U$. By the Leray spectral sequence we conclude the same thing is true for $\mathcal{F}_ i|_ U$ (Cohomology on Sites, Lemma 21.14.6).

Proof of (2). Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \to Y$ be an étale morphism with $V$ affine. Then $V \times _ Y X \to X$ is an étale morphism and $V \times _ Y X$ is a quasi-compact and quasi-separated algebraic space étale over $X$ (details omitted). Hence $H^ p(V \times _ Y X, \mathcal{J})$ is zero by part (1). Since $R^ pf_*\mathcal{J}$ is the sheaf associated to the presheaf $V \mapsto H^ p(V \times _ Y X, \mathcal{J})$ the result is proved. $\square$

Lemma 72.12.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of Noetherian algebraic spaces over $S$. 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 the morphism is quasi-compact and quasi-separated (see Morphisms of Spaces, Lemma 64.8.10). Thus $f_*$ preserve quasi-coherence, see Morphisms of Spaces, Lemma 64.11.2. 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 of Spaces, Proposition 63.32.2), 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 72.6.1. 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 72.6.1 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 case follows from Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) with required vanishing because of Lemma 72.12.4. $\square$

Proposition 72.12.6. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Then the functor (72.5.1.1)

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

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

Proof. Follows immediately from Lemmas 72.12.5 and 72.11.4. $\square$


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