## 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 $Rg_* \circ Rf_* = R(g \circ f)_*$.
$\square$

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.
$\square$

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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