Section 21.20 (08FH): Some properties of K-injective complexes

21.20 Some properties of K-injective complexes

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ . Denote $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ the corresponding localization morphism. 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

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

Similarly, extension by zero $j_! : \textit{Mod}(\mathcal{O}_ U) \to \textit{Mod}(\mathcal{O})$ (see Modules on Sites, Definition 18.19.1) is an exact functor (Modules on Sites, Lemma 18.19.3). Thus it induces a functor

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

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

Lemma 21.20.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ . The restriction of a K-injective complex of $\mathcal{O}$ -modules to $\mathcal{C}/U$ is a K-injective complex of $\mathcal{O}_ U$ -modules.

Proof. Follows immediately from Derived Categories, Lemma 13.31.9 and the fact that the restriction functor has the exact left adjoint $j_!$ . See discussion above. $\square$

Lemma 21.20.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . For $K$ in $D(\mathcal{O})$ we have $H^ p(U, K) = H^ p(\mathcal{C}/U, K|_{\mathcal{C}/U})$ .

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

$H^ q(U, K) = H^ q(\Gamma (U, \mathcal{I}^\bullet )) = H^ q(\Gamma (\mathcal{C}/U, \mathcal{I}^\bullet |_{\mathcal{C}/U}))$

by construction of cohomology. By Lemma 21.20.1 the complex $\mathcal{I}^\bullet |_{\mathcal{C}/U}$ is a K-injective complex representing $K|_{\mathcal{C}/U}$ and the lemma follows. $\square$

Lemma 21.20.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O})$ . The sheafification of

$U \mapsto H^ q(U, K) = H^ q(\mathcal{C}/U, K|_{\mathcal{C}/U})$

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

Proof. The equality $H^ q(U, K) = H^ q(\mathcal{C}/U, K|_{\mathcal{C}/U})$ holds by Lemma 21.20.2. 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 21.20.4. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$ . Given $V \in \mathcal{D}$ , set $U = u(V)$ and denote $g : (\mathcal{C}/U, \mathcal{O}_ U) \to (\mathcal{D}/V, \mathcal{O}_ V)$ the induced morphism of ringed sites (Modules on Sites, Lemma 18.20.1). Then $(Rf_*E)|_{\mathcal{D}/V} = Rg_*(E|_{\mathcal{C}/U})$ for $E$ in $D(\mathcal{O}_\mathcal {C})$ .

Proof. Represent $E$ by a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_\mathcal {C}$ -modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet$ and $Rg_*(E|_{\mathcal{C}/U}) = g_*(\mathcal{I}^\bullet |_{\mathcal{C}/U})$ by Lemma 21.20.1. Since it is clear that $(f_*\mathcal{F})|_{\mathcal{D}/V} = g_*(\mathcal{F}|_{\mathcal{C}/U})$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$ (see Modules on Sites, Lemma 18.20.1 or the more basic Sites, Lemma 7.28.1) the result follows. $\square$

Lemma 21.20.5. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$ . Then $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ as functors $D(\mathcal{O}_\mathcal {C}) \to D(\Gamma (\mathcal{O}_\mathcal {D}))$ . More generally, for $V \in \mathcal{D}$ with $U = u(V)$ we have $R\Gamma (U, -) = R\Gamma (V, -) \circ Rf_*$ .

Proof. Consider the punctual topos $pt$ endowed with $\mathcal{O}_{pt}$ given by the ring $\Gamma (\mathcal{O}_\mathcal {D})$ . There is a canonical morphism $(\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt})$ of ringed topoi inducing the identification on global sections of structure sheaves. Then $D(\mathcal{O}_{pt}) = D(\Gamma (\mathcal{O}_\mathcal {D}))$ . The assertion $R\Gamma (\mathcal{D}, -) \circ Rf_* = R\Gamma (\mathcal{C}, -)$ follows from Lemma 21.19.2 applied to

$(\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D}) \to (pt, \mathcal{O}_{pt})$

The second (more general) statement follows from the first statement after applying Lemma 21.20.4. $\square$

Lemma 21.20.6. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$ . Let $K$ be in $D(\mathcal{O}_\mathcal {C})$ . Then $H^ i(Rf_*K)$ is the sheaf associated to the presheaf

$V \mapsto H^ i(u(V), K) = H^ i(V, Rf_*K)$

Proof. The equality $H^ i(u(V), K) = H^ i(V, Rf_*K)$ follows upon taking cohomology from the second statement in Lemma 21.20.5. Then the statement on sheafification follows from Lemma 21.20.3. $\square$

Lemma 21.20.7. Let $(\mathcal{C}, \mathcal{O}_\mathcal {C})$ be a ringed site. Let $K$ be an object of $D(\mathcal{O}_\mathcal {C})$ and denote $K_{ab}$ its image in $D(\underline{\mathbf{Z}}_\mathcal {C})$ .

There is a canonical map $R\Gamma (\mathcal{C}, K) \to R\Gamma (\mathcal{C}, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$ .
For any $U \in \mathcal{C}$ there is a canonical map $R\Gamma (U, K) \to R\Gamma (U, K_{ab})$ which is an isomorphism in $D(\textit{Ab})$ .
Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. There is a canonical map $Rf_*K \to Rf_*(K_{ab})$ which is an isomorphism in $D(\underline{\mathbf{Z}}_\mathcal {D})$ .

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 $\Gamma (\mathcal{C}, \mathcal{I}^\bullet ) \to \Gamma (\mathcal{C}, \mathcal{J}^\bullet )$ (2) is given by $\Gamma (U, \mathcal{I}^\bullet ) \to \Gamma (U, \mathcal{J}^\bullet )$ and the map in (3) 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 21.20.2 and 21.20.6 it suffices to show that the map

$H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})$

is an isomorphism. Observe that

$H^0(\mathcal{C}, K) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {C})}(\mathcal{O}_\mathcal {C}, K)$

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

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

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

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

Next, we observe that the quasi-isomorphisms $s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal {C}$ with $\mathcal{G}^\bullet$ bounded above complex of flat $\underline{\mathbf{Z}}_\mathcal {C}$ -modules is cofinal in the system. (This follows from Modules on Sites, Lemma 18.28.8 and Derived Categories, Lemma 13.15.4; see discussion in Section 21.17.) Hence we can construct an inverse to the map $H^0(\mathcal{C}, K) \longrightarrow H^0(\mathcal{C}, K_{ab})$ by representing an element $\xi \in H^0(\mathcal{C}, K_{ab})$ by a pair

$(s : \mathcal{G}^\bullet \to \underline{\mathbf{Z}}_\mathcal {C}, a : \mathcal{G}^\bullet \to \mathcal{K}^\bullet )$

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

$(\mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_\mathcal {C}} \mathcal{O}_\mathcal {C} \to \mathcal{O}_\mathcal {C}, \mathcal{G}^\bullet \otimes _{\underline{\mathbf{Z}}_\mathcal {C}} \mathcal{O}_\mathcal {C} \to \mathcal{K}^\bullet )$

The only thing to note here is that the first arrow is a quasi-isomorphism by Lemmas 21.17.12 and 21.17.8. We omit the detailed verification that this construction is indeed an inverse. $\square$

Lemma 21.20.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$ . Denote $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ the corresponding localization morphism. The restriction functor $D(\mathcal{O}) \to D(\mathcal{O}_ U)$ is a right adjoint to extension by zero $j_! : D(\mathcal{O}_ U) \to D(\mathcal{O})$ .

Proof. We have to show that

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!E, F) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E, F|_ U)$

Choose a complex $\mathcal{E}^\bullet$ of $\mathcal{O}_ U$ -modules representing $E$ and choose a K-injective complex $\mathcal{I}^\bullet$ representing $F$ . By Lemma 21.20.1 the complex $\mathcal{I}^\bullet |_ U$ is K-injective as well. Hence we see that the formula above becomes

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(j_!\mathcal{E}^\bullet , \mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{E}^\bullet , \mathcal{I}^\bullet |_ U)$

which holds as $|_ U$ and $j_!$ are adjoint functors (Modules on Sites, Lemma 18.19.2) and Derived Categories, Lemma 13.31.2. $\square$

Lemma 21.20.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . For $L$ in $D(\mathcal{O}_ U)$ and $K$ in $D(\mathcal{O})$ we have $j_!L \otimes _\mathcal {O}^\mathbf {L} K = j_!(L \otimes _{\mathcal{O}_ U}^\mathbf {L} K|_ U)$ .

Proof. Represent $L$ by a complex of $\mathcal{O}_ U$ -modules and $K$ by a K-flat complexe of $\mathcal{O}$ -modules and apply Modules on Sites, Lemma 18.27.9. Details omitted. $\square$

Lemma 21.20.10. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a flat morphism of ringed topoi. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}_\mathcal {C}$ -modules, then $f_*\mathcal{I}^\bullet$ is K-injective as a complex of $\mathcal{O}_\mathcal {D}$ -modules.

Proof. This is true because

$\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {D})}(\mathcal{F}^\bullet , f_*\mathcal{I}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}_\mathcal {C})}(f^*\mathcal{F}^\bullet , \mathcal{I}^\bullet )$

by Modules on Sites, Lemma 18.13.2 and the fact that $f^*$ is exact as $f$ is assumed to be flat. $\square$

Lemma 21.20.11. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be a map of sheaves of rings. If $\mathcal{I}^\bullet$ is a K-injective complex of $\mathcal{O}$ -modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}'$ -modules.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O}')}(\mathcal{G}^\bullet , \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{I}^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{O})}(\mathcal{G}^\bullet , \mathcal{I}^\bullet )$ by Modules on Sites, Lemma 18.27.8. $\square$

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21.20 Some properties of K-injective complexes

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