
13.22 Composition of right derived functors

Sometimes we can compute the right derived functor of a composition. Suppose that $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be left exact functors. Assume that the right derived functors $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$, $RG : D^{+}(\mathcal{B}) \to D^{+}(\mathcal{C})$, and $R(G \circ F) : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{C})$ are everywhere defined. Then there exists a canonical transformation

$t : R(G \circ F) \longrightarrow RG \circ RF$

of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$, see Lemma 13.15.16. This transformation need not always be an isomorphism.

Lemma 13.22.1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be left exact functors. Assume $\mathcal{A}$, $\mathcal{B}$ have enough injectives. The following are equivalent

1. $F(I)$ is right acyclic for $G$ for each injective object $I$ of $\mathcal{A}$, and

2. the canonical map

$t : R(G \circ F) \longrightarrow RG \circ RF.$

is isomorphism of functors of functors from $D^{+}(\mathcal{A})$ to $D^{+}(\mathcal{C})$.

Proof. If (2) holds, then (1) follows by evaluating the isomorphism $t$ on $RF(I) = F(I)$. Conversely, assume (1) holds. Let $A^\bullet$ be a bounded below complex of $\mathcal{A}$. Choose an injective resolution $A^\bullet \to I^\bullet$. The map $t$ is given (see proof of Lemma 13.15.16) by the maps

$R(G \circ F)(A^\bullet ) = (G \circ F)(I^\bullet ) = G(F(I^\bullet ))) \to RG(F(I^\bullet )) = RG(RF(A^\bullet ))$

where the arrow is an isomorphism by Lemma 13.17.7. $\square$

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

$E_2^{p, q} = R^ pG(R^ qF(X))$

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

Proof. We may represent $X$ by a bounded below complex $A^\bullet$. Choose an injective resolution $A^\bullet \to I^\bullet$. Choose a Cartan-Eilenberg resolution $F(I^\bullet ) \to I^{\bullet , \bullet }$ using Lemma 13.21.2. Apply the second spectral sequence of Lemma 13.21.3. $\square$

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