The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

20.30 Some properties of K-injective complexes

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $U \subset X$ be an open subset. Denote $j : (U, \mathcal{O}_ U) \to (X, \mathcal{O}_ X)$ the corresponding open immersion. The pullback functor $j^*$ is exact as it is just the restriction functor. Thus derived pullback $Lj^*$ is computed on any complex by simply restricting the complex. We often simply denote the corresponding functor

\[ D(\mathcal{O}_ X) \to D(\mathcal{O}_ U), \quad E \mapsto j^*E = E|_ U \]

Similarly, extension by zero $j_! : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O}_ X)$ (see Sheaves, Section 6.31) is an exact functor (Modules, Lemma 17.3.4). Thus it induces a functor

\[ j_! : D(\mathcal{O}_ U) \to D(\mathcal{O}_ X),\quad F \mapsto j_!F \]

by simply applying $j_!$ to any complex representing the object $F$.

Lemma 20.30.1. Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. The restriction of a K-injective complex of $\mathcal{O}_ X$-modules to $U$ is a K-injective complex of $\mathcal{O}_ U$-modules.

Proof. Follows from Derived Categories, Lemma 13.29.9 and the fact that the restriction functor has the exact left adjoint $j_!$. For the construction of $j_!$ see Sheaves, Section 6.31 and for exactness see Modules, Lemma 17.3.4. $\square$

Lemma 20.30.2. Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. For $K$ in $D(\mathcal{O}_ X)$ we have $H^ p(U, K) = H^ p(U, K|_ U)$.

Proof. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules representing $K$. Then

\[ H^ q(U, K) = H^ q(\Gamma (U, \mathcal{I}^\bullet )) = H^ q(\Gamma (U, \mathcal{I}^\bullet |_ U)) \]

by construction of cohomology. By Lemma 20.30.1 the complex $\mathcal{I}^\bullet |_ U$ is a K-injective complex representing $K|_ U$ and the lemma follows. $\square$

Lemma 20.30.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_ X)$. The sheafification of

\[ U \mapsto H^ q(U, K) = H^ q(U, K|_ U) \]

is the $q$th cohomology sheaf $H^ q(K)$ of $K$.

Proof. The equality $H^ q(U, K) = H^ q(U, K|_ U)$ holds by Lemma 20.30.2. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $K$. Then

\[ H^ q(U, K) = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ q(U) \to \mathcal{I}^{q + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{q - 1}(U) \to \mathcal{I}^ q(U))}. \]

by our construction of cohomology. Since $H^ q(K) = \mathop{\mathrm{Ker}}(\mathcal{I}^ q \to \mathcal{I}^{q + 1})/ \mathop{\mathrm{Im}}(\mathcal{I}^{q - 1} \to \mathcal{I}^ q)$ the result is clear. $\square$

Lemma 20.30.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Given an open subspace $V \subset Y$, set $U = f^{-1}(V)$ and denote $g : U \to V$ the induced morphism. Then $(Rf_*E)|_ V = Rg_*(E|_ U)$ for $E$ in $D(\mathcal{O}_ X)$.

Proof. Represent $E$ by a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X$-modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet $ and $Rg_*(E|_ U) = g_*(\mathcal{I}^\bullet |_ U)$ by Lemma 20.30.1. Since it is clear that $(f_*\mathcal{F})|_ V = g_*(\mathcal{F}|_ U)$ for any sheaf $\mathcal{F}$ on $X$ the result follows. $\square$

Lemma 20.30.5. Let $f : X \to Y$ be a morphism of ringed spaces. Then $R\Gamma (Y, -) \circ Rf_* = R\Gamma (X, -)$ as functors $D(\mathcal{O}_ X) \to D(\Gamma (Y, \mathcal{O}_ Y))$. More generally for $V \subset Y$ open and $U = f^{-1}(V)$ we have $R\Gamma (U, -) = R\Gamma (V, -) \circ Rf_*$.

Proof. Let $Z$ be the ringed space consisting of a singleton space with $\Gamma (Z, \mathcal{O}_ Z) = \Gamma (Y, \mathcal{O}_ Y)$. There is a canonical morphism $Y \to Z$ of ringed spaces inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_ Z) = D(\Gamma (Y, \mathcal{O}_ Y))$. Hence the assertion $R\Gamma (Y, -) \circ Rf_* = R\Gamma (X, -)$ follows from Lemma 20.29.2 applied to $X \to Y \to Z$.

The second (more general) statement follows from the first statement after applying Lemma 20.30.4. $\square$

Lemma 20.30.6. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $K$ be in $D(\mathcal{O}_ X)$. Then $H^ i(Rf_*K)$ is the sheaf associated to the presheaf

\[ V \mapsto H^ i(f^{-1}(V), K) = H^ i(V, Rf_*K) \]

Proof. The equality $H^ i(f^{-1}(V), K) = H^ i(V, Rf_*K)$ follows upon taking cohomology from the second statement in Lemma 20.30.5. Then the statement on sheafification follows from Lemma 20.30.3. $\square$

Lemma 20.30.7. Let $X$ be a ringed space. Let $K$ be an object of $D(\mathcal{O}_ X)$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_ X)$.

  1. For any open $U \subset X$ there is a canonical map $R\Gamma (U, K) \to R\Gamma (U, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$.

  2. Let $f : X \to Y$ be a morphism of ringed spaces. There is a canonical map $Rf_*K \to Rf_*(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_ Y)$.

Proof. The map is constructed as follows. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $K$. Choose a quasi-isomorpism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ where $\mathcal{J}^\bullet $ is a K-injective complex of abelian groups. Then the map in (1) is given by $\Gamma (U, \mathcal{I}^\bullet ) \to \Gamma (U, \mathcal{J}^\bullet )$ and the map in (2) is given by $f_*\mathcal{I}^\bullet \to f_*\mathcal{J}^\bullet $. To show that these maps are isomorphisms, it suffices to prove they induce isomorphisms on cohomology groups and cohomology sheaves. By Lemmas 20.30.2 and 20.30.6 it suffices to show that the map

\[ H^0(X, K) \longrightarrow H^0(X, K_{ab}) \]

is an isomorphism. Observe that

\[ H^0(X, K) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{O}_ X, K) \]

and similarly for the other group. Choose any complex $\mathcal{K}^\bullet $ of $\mathcal{O}_ X$-modules representing $K$. By construction of the derived category as a localization we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\mathcal{O}_ X, K) = \mathop{\mathrm{colim}}\nolimits _{s : \mathcal{F}^\bullet \to \mathcal{O}_ X} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(\mathcal{F}^\bullet , \mathcal{K}^\bullet ) \]

where the colimit is over quasi-isomorphisms $s$ of complexes of $\mathcal{O}_ X$-modules. Similarly, we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\underline{\mathbf{Z}}_ X)}(\underline{\mathbf{Z}}_ X, K) = \mathop{\mathrm{colim}}\nolimits _{s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_ X} \mathop{\mathrm{Hom}}\nolimits _{K(\underline{\mathbf{Z}}_ X)}(\mathcal{G}^\bullet , \mathcal{K}^\bullet ) \]

Next, we observe that the quasi-isomorphisms $s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_ X$ with $\mathcal{G}^\bullet $ bounded above complex of flat $\underline{\mathbf{Z}}_ X$-modules is cofinal in the system. (This follows from Modules, Lemma 17.16.6 and Derived Categories, Lemma 13.16.5; see discussion in Section 20.27.) Hence we can construct an inverse to the map $H^0(X, K) \longrightarrow H^0(X, K_{ab})$ by representing an element $\xi \in H^0(X, K_{ab})$ by a pair

\[ (s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_ X, a : \mathcal{G}^\bullet \to \mathcal{K}^\bullet ) \]

with $\mathcal{G}^\bullet $ a bounded above complex of flat $\underline{\mathbf{Z}}_ X$-modules and sending this to

\[ (\mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_ X} \mathcal{O}_ X \to \mathcal{O}_ X, \mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_ X} \mathcal{O}_ X \to \mathcal{K}^\bullet ) \]

The only thing to note here is that the first arrow is a quasi-isomorphism by Lemmas 20.27.12 and 20.27.8. We omit the detailed verification that this construction is indeed an inverse. $\square$

Lemma 20.30.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $U \subset X$ be an open subset. Denote $j : (U, \mathcal{O}_ U) \to (X, \mathcal{O}_ X)$ the corresponding open immersion. The restriction functor $D(\mathcal{O}_ X) \to D(\mathcal{O}_ U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_ U) \to D(\mathcal{O}_ X)$.

Proof. This follows formally from the fact that $j_!$ and $j^*$ are adjoint and exact (and hence $Lj_! = j_!$ and $Rj^* = j^*$ exist), see Derived Categories, Lemma 13.28.5. $\square$

Lemma 20.30.9. Let $f : X \to Y$ be a flat morphism of ringed spaces. If $\mathcal{I}^\bullet $ is a K-injective complex of $\mathcal{O}_ X$-modules, then $f_*\mathcal{I}^\bullet $ is K-injective as a complex of $\mathcal{O}_ Y$-modules.

Proof. This is true because

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ Y)}(\mathcal{F}^\bullet , f_*\mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_ X)}(f^*\mathcal{F}^\bullet , \mathcal{I}^\bullet ) \]

by Sheaves, Lemma 6.26.2 and the fact that $f^*$ is exact as $f$ is assumed to be flat. $\square$


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