The Stacks project

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20.29 Cohomology of unbounded complexes

Let $(X, \mathcal{O}_ X)$ be a ringed space. The category $\textit{Mod}(\mathcal{O}_ X)$ is a Grothendieck abelian category: it has all colimits, filtered colimits are exact, and it has a generator, namely

\[ \bigoplus \nolimits _{U \subset X\text{ open}} j_{U!}\mathcal{O}_ U, \]

see Modules, Section 17.3 and Lemmas 17.16.5 and 17.16.6. By Injectives, Theorem 19.12.6 for every complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules there exists an injective quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $ to a K-injective complex of $\mathcal{O}_ X$-modules. Hence we can define

\[ R\Gamma (X, \mathcal{F}^\bullet ) = \Gamma (X, \mathcal{I}^\bullet ) \]

and similarly for any left exact functor, see Derived Categories, Lemma 13.29.7. For any morphism of ringed spaces $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ we obtain

\[ Rf_* : D(X) \longrightarrow D(Y) \]

on the unbounded derived categories.

Lemma 20.29.1. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The functor $Rf_*$ defined above and the functor $Lf^*$ defined in Lemma 20.28.1 are adjoint:

\[ \mathop{\mathrm{Hom}}\nolimits _{D(X)}(Lf^*\mathcal{G}^\bullet , \mathcal{F}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(Y)}(\mathcal{G}^\bullet , Rf_*\mathcal{F}^\bullet ) \]

bifunctorially in $\mathcal{F}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(X))$ and $\mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(Y))$.

Proof. This follows formally from the fact that $Rf_*$ and $Lf^*$ exist, see Derived Categories, Lemma 13.28.5. $\square$

Lemma 20.29.2. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_ Z)$.

Proof. By Lemma 20.29.1 we see that $Rg_* \circ Rf_*$ is adjoint to $Lf^* \circ Lg^*$. We have $Lf^* \circ Lg^* = L(g \circ f)^*$ by Lemma 20.28.2 and hence by uniqueness of adjoint functors we have $Rg_* \circ Rf_* = R(g \circ f)_*$. $\square$

Remark 20.29.3. The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

is a commutative diagram of ringed spaces. Let $K$ be an object of $D(\mathcal{O}_ X)$. Then there exists a canonical base change map

\[ Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K \]

in $D(\mathcal{O}_{S'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \to L(g')^*K$ Since $L(f')^*Lg^* = L(g')^*Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \to K$.

Remark 20.29.4. Consider a commutative diagram

\[ \xymatrix{ X' \ar[r]_ k \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ l \ar[d]_{g'} & Y \ar[d]^ g \\ Z' \ar[r]^ m & Z } \]

of ringed spaces. Then the base change maps of Remark 20.29.3 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition

\begin{align*} Lm^* \circ R(g \circ f)_* & = Lm^* \circ Rg_* \circ Rf_* \\ & \to Rg'_* \circ Ll^* \circ Rf_* \\ & \to Rg'_* \circ Rf'_* \circ Lk^* \\ & = R(g' \circ f')_* \circ Lk^* \end{align*}

is the base change map for the rectangle. We omit the verification.

Remark 20.29.5. Consider a commutative diagram

\[ \xymatrix{ X'' \ar[r]_{g'} \ar[d]_{f''} & X' \ar[r]_ g \ar[d]_{f'} & X \ar[d]^ f \\ Y'' \ar[r]^{h'} & Y' \ar[r]^ h & Y } \]

of ringed spaces. Then the base change maps of Remark 20.29.3 for the two squares compose to give the base change map for the outer rectangle. More precisely, the composition

\begin{align*} L(h \circ h')^* \circ Rf_* & = L(h')^* \circ Lh_* \circ Rf_* \\ & \to L(h')^* \circ Rf'_* \circ Lg^* \\ & \to Rf''_* \circ L(g')^* \circ Lg^* \\ & = Rf”_* \circ L(g \circ g')^* \end{align*}

is the base change map for the rectangle. We omit the verification.

Remark 20.29.6. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product

\[ Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \longrightarrow Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \]

in $D(\mathcal{O}_ Y)$ for all $K, L$ in $D(\mathcal{O}_ X)$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) = Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L$ (Lemma 20.28.3) with the map $Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.

Lemma 20.29.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. There exists a canonical spectral sequence $(E_ r, \text{d}_ r)_{r \geq 1}$ of bigraded $\Gamma (X, \mathcal{O}_ X)$-modules with $d_ r$ of bidegree $(r, -r + 1)$ and

\[ E_1^{p, q} = H^{p + q}(X, \text{gr}^ p\mathcal{F}^\bullet ) \]

If for every $n$ we have

\[ H^ n(X, F^ p\mathcal{F}^\bullet ) = 0\text{ for }p \gg 0 \quad \text{and}\quad H^ n(X, F^ p\mathcal{F}^\bullet ) = H^ n(X, \mathcal{F}^\bullet )\text{ for }p \ll 0 \]

then the spectral sequence is bounded and converges to $H^*(X, \mathcal{F}^\bullet )$.

Proof. (For a proof in case the complex is a bounded below complex of modules with finite filtrations, see the remark below.) Choose an map of filtered complexes $j : \mathcal{F}^\bullet \to \mathcal{J}^\bullet $ as in Injectives, Lemma 19.13.7. The spectral sequence is the spectral sequence of Homology, Section 12.21 associated to the filtered complex

\[ \Gamma (X, \mathcal{J}^\bullet ) \quad \text{with}\quad F^ p\Gamma (X, \mathcal{J}^\bullet ) = \Gamma (X, F^ p\mathcal{J}^\bullet ) \]

Since cohomology is computed by evaluating on K-injective representatives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma 12.21.13. $\square$

Remark 20.29.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. If $\mathcal{F}^\bullet $ is bounded from below and for each $n$ the filtration on $\mathcal{F}^ n$ is finite, then there is a construction of the spectral sequence in Lemma 20.29.7 avoiding Injectives, Lemma 19.13.7. Namely, by Derived Categories, Lemma 13.26.9 there is a filtered quasi-isomorphism $i : \mathcal{F}^\bullet \to \mathcal{I}^\bullet $ of filtered complexes with $\mathcal{I}^\bullet $ bounded below, the filtration on $\mathcal{I}^ n$ is finite for all $n$, and with each $\text{gr}^ p\mathcal{I}^ n$ an injective $\mathcal{O}_ X$-module. Then we take the spectral sequence associated to

\[ \Gamma (X, \mathcal{I}^\bullet ) \quad \text{with}\quad F^ p\Gamma (X, \mathcal{I}^\bullet ) = \Gamma (X, F^ p\mathcal{I}^\bullet ) \]

Since cohomology can be computed by evaluating on bounded below complexes of injectives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma 12.21.11. In fact, this is a special case of the spectral sequence in Derived Categories, Lemma 13.26.14.

Example 20.29.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We can apply Lemma 20.29.7 with $F^ p\mathcal{F}^\bullet = \tau _{\leq -p}\mathcal{F}^\bullet $. (If $\mathcal{F}^\bullet $ is bounded below we can use Remark 20.29.8.) Then we get a spectral sequence

\[ E_1^{p, q} = H^{p + q}(X, H^{-p}(\mathcal{F}^\bullet )[p]) = H^{2p + q}(X, H^{-p}(\mathcal{F}^\bullet )) \]

After renumbering $p = -j$ and $q = i + 2j$ we find that for any $K \in D(\mathcal{O}_ X)$ there is a spectral sequence $(E'_ r, d'_ r)_{r \geq 2}$ of bigraded modules with $d'_ r$ of bidegree $(r, -r + 1)$, with

\[ (E'_2)^{i, j} = H^ i(X, H^ j(K)) \]

If $K$ is bounded below (for example), then this spectral sequence is bounded and converges to $H^{i + j}(X, K)$. In the bounded below case this spectral sequence is an example of the second spectral sequence of Derived Categories, Lemma 13.21.3 (constructed using Cartan-Eilenberg resolutions).


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