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Recently I saw the following fact:

Proposition (Hartshorne III.1.2A): Let $\mathscr{A,B}$ be abelian categories and suppose $\mathscr{A}$ has enough injectives. Let $F:\mathscr{A}\to \mathscr{B}$ a covariant left exact functor. Suppose there is an exact sequence $$0\to A\to J^0\to J^1\to \cdots$$ where each $J^i$ is acyclic for $F$, $i\geq 0$. Then for each $i\geq 0$ there is a natural isomorphism $R^iF(A)\cong h^i(F(J^\bullet))$.

This says that the derived functors $R^iF$ can be obtained as the homology of an $F$-acyclic resolution. How can I prove this using basic homological algebra? Are there references to proofs of this fact?

KReiser
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user9490170
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  • Perhaps https://math.stackexchange.com/questions/1038292/why-can-we-use-flabby-sheaves-to-define-cohomology/1038346#1038346 is helpful. – Hanno Jan 14 '20 at 12:54
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    split the long sequence J to short sequence ,then obtain a long sequence for every short sequence , then just track RiF(A) – user9490170 Jan 14 '20 at 13:40
  • A good reference for homological algebra and derived functors is the book of Gelfand and Manin : https://www.maths.ed.ac.uk/~v1ranick/papers/gelfmani.pdf – Dominique Mattei Jan 15 '20 at 16:41

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