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

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

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