The total derived functor of a composition is the composition of the total derived functors.

Lemma 20.13.7. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. In this case $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors from $D^{+}(X) \to D^{+}(Z)$.

Proof. We are going to apply Derived Categories, Lemma 13.22.1. It is clear that $g_* \circ f_* = (g \circ f)_*$, see Sheaves, Lemma 6.21.2. It remains to show that $f_*\mathcal{I}$ is $g_*$-acyclic. This follows from Lemma 20.11.10 and the description of the higher direct images $R^ ig_*$ in Lemma 20.7.3. $\square$

Comment #1228 by David Corwin on

Suggested slogan: The total derived functor of a composition is the composition of the total derived functors

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).