Proposition 63.8.2. In the situation above, we have

$\text{gr}^ p \circ RT = RT \circ \text{gr}^ p$

where the $RT$ on the left is the filtered derived functor while the one on the right is the total derived functor. That is, there is a commuting diagram

$\xymatrix{ DF^+(\mathcal{A}) \ar[r]^{RT} \ar[d]_{\text{gr}^ p} & DF^+(\mathcal{B}) \ar[d]^{\text{gr}^ p}\\ D^+(\mathcal{A}) \ar[r]^{RT} & D^+(\mathcal{B}).}$

Proof. Omitted. $\square$

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