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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01F5. Beware of the difference between the letter 'O' and the digit '0'.