(acyclic)
Let \(\mca\) be an abelian category with enough injectives and \(F: \mca \to \mcb\) a left exact additive functor, so that the \(R^iF\) exist. An object \(A \in \mca\) is \(F\)-acyclic if the higher right derived functors vanish: \[ R^iF(A) = 0 \text{ for all } i > 0 . \] The same definition applies to left derived functors and to contravariant \(F\).
If \(0\to A\to L^\bullet\) is a resolution of \(A\) by \(F\)-acyclic objects, then \(R^iF(A)\cong h^i(F(L^\bullet))\) for all \(i\ge0\) [@Har10a, Proposition III.1.2A].
Injective objects are \(F\)-acyclic for every left exact \(F\), and projective objects are acyclic for the left derived functors of every right exact functor. An acyclic object need not be injective: on a one-point space every sheaf is \(\Gamma\)-acyclic, and the constant sheaf \(\ZZ\) is not an injective abelian group.
Flasque sheaves are \(\Gamma\)-acyclic, so a flasque resolution computes \(H^i(X;\mcf)\). Free and projective modules are acyclic for \(\wait \tensor_A N\), so a free resolution computes \(\Tor\). Quasicoherent sheaves on an affine scheme are \(\Gamma\)-acyclic by Serre’s theorem; hence for a Noetherian separated scheme \(X\), an open affine cover \(\mathfrak U\), and a quasicoherent sheaf \(\mcf\), the Čech complex of \(\mathfrak U\) computes \(H^i(X,\mcf)\) [@Har10a, Theorem III.4.5].
Dependencies
D-DEFDERIVDerived functorsuses