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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