Lemma 13.22.2 (Grothendieck spectral sequence). With assumptions as in Lemma 13.22.1 and assuming the equivalent conditions (1) and (2) hold. Let $X$ be an object of $D^{+}(\mathcal{A})$. There exists a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ consisting of bigraded objects $E_ r$ of $\mathcal{C}$ with $d_ r$ of bidegree $(r, - r + 1)$ and with

$E_2^{p, q} = R^ pG(H^ q(RF(X)))$

Moreover, this spectral sequence is bounded, converges to $H^*(R(G \circ F)(X))$, and induces a finite filtration on each $H^ n(R(G \circ F)(X))$.

Proof. We may represent $X$ by a bounded below complex $A^\bullet$. Choose an injective resolution $A^\bullet \to I^\bullet$. Choose a Cartan-Eilenberg resolution $F(I^\bullet ) \to I^{\bullet , \bullet }$ using Lemma 13.21.2. Apply the second spectral sequence of Lemma 13.21.3. $\square$

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