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The Stacks project

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 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 13.14.16) 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 13.16.7. \square


Comments (2)

Comment #5387 by Will Chen on

In (2), there's an extra "of functors".

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  • 4 comment(s) on Section 13.22: Composition of right derived functors

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